Mathematical Physics, Analysis and Geometry - Volume 12

Math Phys Anal Geom (2009) 12:1–18 DOI 10.1007/s11040-008-9049-4

A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence Vladimir N. Grebenev · Martin Oberlack

Received: 25 February 2008 / Accepted: 22 October 2008 / Published online: 20 November 2008 © Springer Science + Business Media B.V. 2008

Abstract We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Kármán-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Kármán-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Kármán-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals −1.

V. N. Grebenev (B) Institute of Computational Technologies, Russian Academy of Science, Lavrentjev ave. 6, Novosibirsk 630090, Russia e-mail: [email protected] M. Oberlack Fluid Dynamics, Technische Universität Darmstadt, Hochschulstrasse 1, Darmstadt 64289, Germany e-mail: [email protected]

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V.N. Grebenev, M. Oberlack

Keywords Beltrami surface · Closure model for the von Kármán-Howarth equation · Homogeneous isotropic turbulence · Riemannian metric · Two-point correlation tensor · Length scales of turbulent motion Mathematics Subject Classifications (2000) 76F05 · 76F55 · 53B21 · 53B50 · 58J70

1 Introduction Turbulent fluid dynamics is characterized by ranking turbulent motions in size from scales ∼ l of the flow under consideration to much smaller scales which become progressively smaller as the Reynolds number increases. One of a fundamental problem of turbulent fluid dynamics consist of studying the shape dynamics of a fluid volume. The first concept in Richardson point of view is that the turbulence can be considered to compose eddies (a turbulent motion localized within a region of size l) of different sizes. Richardson’s notion is that the eddies are evolved in time, transferring their energy to smaller scale motions. These smaller eddies undergo a similar cascade process, and transfer their energy to yet smaller eddies in the inertial range and so on—continuous until the Reynolds number is sufficiently small that molecular viscosity is effective in dissipating the kinetic energy. The characteristic features of turbulence—its distribution of eddy sizes, shapes, speeds, vorticity, circulation, and viscous dissipation—may all be captured within the statistical approach to fully developed turbulence, and several questions can be posed. What are the sizes of the eddies which are generated in Richardson scenario? As time increases, how the shape of eddies is deformed? While there are many efforts in this direction, the aim of this paper is to present an approach that is based on the use of methods of Riemannian geometry for studying the shape dynamics of eddies, in particular, on the interaction between the deformation of geometric quantities (shape form, curvature and other) of a manifold (a singled out fluid volume) equipped with a family of Riemannian metrics (length scales of turbulent motion) and the deformation of these Riemannian metrics in time t. Our approach is conceptually similar to the Ricci flow ideas [1]. The Ricci flow is an evolution differential equation on the space of Riemannian metrics, the behavior of smooth Riemannian metrics which evolves under the flow may serve as a model to tell us something about the geometry of an underlying manifold. The advantage of this approach is that we can control the deformation of geometric quantities of the manifold under consideration in time, and often a Ricci flow deforms an initial metric to a canonical metric and a key point is to control the so-called injectivity radius of the metrics. A well-known example of the above is a Ricci flow that is starting from a round sphere S N with an initial metric gmn (x, 0) = g(0) such that Rmn = λgmn (x, 0), λ ∈ R where Rmn is the Ricci tensor. This metric is known as

Interpretation of the Second-Order Structure Function

3

Einstein metric. The sphere shrinks homothetically flow to a point in a finite time under the Ricci flow ∂ gmn = −2Rmn , ∂t where the evolving metrics are given by the formula gmn (x, t) = ρ 2 (t)gmn (0) ≡ (1 − 2λt)gmn (0),

λ = N − 1,

and the sphere shrinks homothetically to a point as t → T = 1/2(N − 1). Another example of this type would be if g0 is a hyperbolic metric or an Einstein metric of negative scalar curvature. Then the manifold will expand homothetically for all times. Indeed if Rmn = −λgmn (x, 0) then ρ(t) satisfies dρ λ = , dt ρ with the solution ρ 2 (t) = 1 + 2λt. Hence the evolving metrics gmn (x, t) = ρ 2 (t)gmn (x, 0) exists and expands homothetically for all times. These illustrative examples give us a feeling how the Ricci flow can deform a manifold equipped with an initial Riemannian metric g(0). In the general case, the Ricci flow behaves more wildly. In this paper, we deal with homogeneous isotropic turbulence and emphasis is placed on the use of the specific form of a closure model [2, 3] for the von Kármán-Howarth equation [4] to introduce into consideration a family of Riemannian metrics. Inspired by the Ricci flow idea, we study the behavior of Riemannian metrics constructed and as a consequence, the deformation of some metric quantities of an underlying Riemannian manifold can be determined. In order to equip a model manifold (a singled out fluid volume within turbulent flow) by a family of Riemannian metrics (length scales of turbulent motion), we rewrite this model in the form of an evolution equation and show that the right-hand side of this evolution equation coincides with the so-called radial part of a Laplace-Beltrami type operator. This enables to construct Riemannian metrics (length scales of turbulent motion) compatible with the specific form of this closure model. We recall that the Laplace-Beltrami operator contains a metric tensor of a Riemannian manifold where this operator is defined on. This is a crucial peculiarity of this operator that makes its possible to investigate geometric characteristics of an underlying Riemannian manifold. Using the selfsimilar solution obtained of the closure model for the von Kármán-Howarth equation under consideration, we calculate explicitly the deformation of this family of metrics in time. As a remarkable fact, we note that the above-mentioned selfsimilar solution coincides in the new variables with the element of Beltrami surface (or pseudo-sphere). This is a canonical surface of the constant (sectional) curvature equals −1 [5]. Examining length scales of turbulence motion, we can see that some scales analyzed are based on the use of Euclidian metric to measure a distance.

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However, it is not so clear why we use Euclidian metric in turbulence to define a length scale of turbulent motion without taking into account the geometry of turbulent pattern. The well-known example, where we need a correction of (linear) length scale, is the use of Prandtl’s mixing-length scale lm [6] in the problem of decaying fluid oscillations near a wall. In this problem, a modification of Prandtl’s mixing-length scale is taken in the following (nonlinear) form: lm = κr(1 − exp(−r/A)) [6]. The length scale lm plays the role of a measure of the transversal displacement of fluid particles under turbulent fluctuations. Although the above example comes from the theory of wall turbulent flows, nevertheless this fact reflects understanding to make a correction of some (linear) length scales. We note that even in the case of homogeneous isotropic turbulence there is a relatively small number of publications devoted to numerical modeling isotropic homogeneous turbulence [7] and there are very few results devoted to mathematical analysis of the von Kármán-Howarth equation for the isotropic two-point correlation function. We only mention here the paper [8] wherein this unclosed equation was studied in the framework of the group classification problem of differential equations [9]. We do not discuss the details of the Kolmogorov theory (which tell us that the statistical properties of small scales depend only on the mean rate of energy dissipation  and the correlation distance r) but remark, however, that still are many discussions on whether small scale fluctuations are isotropic or not and that the Richardson scenario may not be valid. Consequently, the velocity statistics in the inertial sub-range may have nonuniversal features. The notion of intermittency is attributed to the violation of local homogeneity of turbulence. This phenomenon leads to the anomalous scaling and reflects a symmetry breaking in the case of ν → 0. From a physical point of view as the viscosity tends to zero turbulence become highly intermittent, and vorticity is concentrated on sets of a small measure and scenario of turbulent motion is complicated significantly. Here we do not review the papers based on the methods of Lagrangian formalism (i.e. the description of turbulent motion of fluids particles) for the stochastic description of turbulence since our approach lies in another field of mathematical investigations of this phenomenon. The difference between the application of Lagrangian formalism method for turbulence (exhaustive reviews on this topic can be found in [10, 11]) and the approach presented here is the same as using Lagrangian and Euler variables in hydrodynamics. We do not look at how a marked fluid particle or an ensemble of marked fluid particles (the separation distance between marked particles) is traveled in turbulent flow but we prefer to observe entirely the deformation of length scales of turbulent motion localized within a singled out fluid volume of this flow in time. The paper is organized as follows. Section 2 is devoted to a closure model for the von Kármán-Howarth equation. Observe that this model holds (see [2]) for a wide range of well accepted turbulence theories for homogeneous isotropic turbulence as there is Kolmogorov first and second similarity hypothesis. In Section 3, we show how to equip a model manifold (a singled out fluid

Interpretation of the Second-Order Structure Function

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volume) by a family of Riemannian metrics (length scales of turbulent motion) exploring the specific form of the above-mentioned closure model for the von Kármán-Howarth equation limited to sufficiently large Reynolds numbers. Moreover, we give a geometric interpretation of the second-order structure function D LL . At the end of Section 3 we present the results [12] of group analysis of the von Kármán-Howarth equation (in its inviscid form) and indicate two scaling symmetries admitted by this equation that enable us to find a whole class of selfsimilar solutions. We show that one implicit selfsimilar solution, which corresponds to Loitsyansky decay low [13], coincides (in the new variables) with the element of Beltrami surface (or pseudosphere). Negativity of the curvature of Beltarmi surface means a stochastic behavior of geodesic curves located on this surface [14]. As was noted by Arnold [14], this property leads to the so-called exponential instability of the geodesic flow. Here we do not develop this topic. Appendix includes a formal derivation of the closure relationship [2] (the algebraic approximation for the triple correlation function) for the von Kármán-Howarth equation limited to sufficiently large Reynolds numbers in the framework of the method of differential constraints [15]. In concluding remarks, we provide the results obtained by physical comments to some extent.

2 Closed Model for the Von Kármán-Howarth Equation We begin with basic notions of homogeneous isotropic turbulence. 2.1 Two-Point Velocity Correlation Tensor Traditional Eulerian turbulence models employ the Reynolds decomposition to separate the fluid velocity u at a point x into its mean and fluctuating components as u = u¯ + u where the symbol (¯·) denotes the Eulerian mean sometimes also called Reynolds averaging. In particular, the concept of two- and multi-point correlation functions was born out of the necessity to obtain length-scale information on turbulent flows. At the same time the resulting correlation equations have considerably less unknown terms at the expense of additional dimensions in the equations. In each of the correlation equations of tensor order n an additional tensor of the order n + 1 appears as unknown term, see for details [16]. The first of the infinite sequence of correlation functions is the two-point correlation tensor defined as Bij(x, x ; tc ) = (ui (x; tc ) − ui (x ; tc ))(uj(x; tc ) − uj(x ; tc )),

(2.1)

where u (x; tc ) and u (x ; tc ) are fluctuating velocities at the points (x; tc ) and (x ; tc ) for each fixed tc ∈ R+ . Therefore, Bij(x, x ; t) defines a tensor field of the independent variables x, x and t on a domain D of the Euclidian space R+ × R6 .

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The assumption of isotropy and homogeneity of a turbulent flow (invariance with respect to rotation, reflection and translation) implies that this tensor may be written in the form [13] Bij(r, tc ) = ui (x; tc )uj(x + r; tc ),

(2.2)

which acts in the so-called correlation space K3 ≡ {r = (r1 , r2 , r3 )}, K3  R3 for each tc , where r = x − x . Moreover, for isotropic turbulence Bij(r, tc ) is a symmetric tensor which depends only on the length |r| of the vector r = r(x, x , tc ), (x, x ) ∈ R6 , and the correlations Bij can be expressed by using only the longitudinal correlational function B LL (|r|, tc ) and the transversal correlation function B N N (|r|, tc ). 2.2 Closure of the Von Kárman-Howarth Equation The correlation functions directly connect the concept of length scales with the result of an actual flow measurement. However, the two-point correlation functions yield no information on the energy, that is contained in a given interval of separation r. The third-order correlations function B LL,L provides information about the energy fluxes between scales. The von KármánHowarth equation relates the time derivative of the component B LL of the two-point correlation tensor to the divergences of the third-order correlation function B LL,L and has the following form   ∂ B LL ∂ B LL 1 ∂ 4 = 4 r B LL,L + 2ν , ∂t r ∂r ∂r

(2.3)

where ν is the kinematic viscosity coefficient, r = |r|. This equation directly follows from the Navier-Stokes equations [13]. Originally, the invariance theory of isotropic turbulence was introduced by von Kármán and Howarth [4] and refined by Robertson [17], who reviewed this equation in the light of classical tensor invariant theory. Arad, L’vov and Procaccia [18] extended these fundamental results by considering projections of the fluid velocity correlation dynamics onto irreducible representation of the SO(N) symmetry group. Equation 2.3 is not closed since it contains two unknowns B LL and B LL,L which cannot be defined from (2.3) alone without the use of additional hypotheses. The simplest assumption is the Kármán–Howarth hypothesis on the similarity of the correlation functions B LL and B LL,L which is B LL (r, t) = u2 (t) f (η), B LL,L (r, t) = (u2 (t))3/2 h(η),

η = r/L(t),

(2.4)

where u2 (t) is the velocity scale for the turbulent kinetic energy, (u2 (t))3/2 is the scale for the turbulent transfer and L(t) is a single global length scale of the turbulence. Substituting these hypothesized expressions into Eq. 2.3, it is

Interpretation of the Second-Order Structure Function

7

straightforward to demonstrate that this equation admits a complete similarity solution of type (2.4) only when the Reynolds number Re = u2 (t)1/2 L(t)/ν is a finite constant. In fact, this directly relates u2 (t) to L(t). It is known that this condition is normally not satisfied in experimental measurements of decaying isotropic turbulence at finite Reynolds numbers. Batchelor and Townsend [19] carried out a similarity analysis of this problem in Fourier space and showed that a similarity solution under this constraint could be found during the final period of decay when the nonlinear terms become negligible. Millionshchikov in [20] outlined a more general hypotheses which produces parametric models of isotropic turbulence based on a closure procedure for von the Kármán–Howarth equation. The essence of these hypotheses is that B LL,L is given by the following relation of gradient-type B LL,L = 2K

∂ B LL , ∂r

(2.5)

where K has the dimension of the turbulent kinematic viscosity which is characterized by a single length and velocity scale. Millionshchikov’s hypotheses [20] assumes that K = κ1 u2

1/2

r,

u2 = B LL (0, t),

(2.6)

where κ1 denotes an empirical constant. An initial-boundary value problem for the Millionshtchikov closure model has been studied in [21] wherein the theory of contractive semigroups was applied to find a solution to the problem by the use of a Chorin-type formula. A way of closing the von Kármán–Howarth equation was suggested by Oberlack in [2] which connects the two-point correlation functions of the thirdorder B LL,L and the second order B LL by using the gradient type hypothesis, that according to [2, 3] takes the form K=

1/2 κ2r D LL ,

D LL = 2[u2 − κ0 B LL (r, t)],

√ 2 κ0 = 1, κ2 = , (2.7) 5C3/2

where C is the Kolmogorov constant. The Millionshchikov hypotheses is a consequence of the above formula in the case of κ0 = 0. Comparison with experimental data was done calculating the triple correlation h (the normalized triple-correlation function) out of measured values of the normalized double correlation function f using the model (2.5), (2.7). The normalized double correlation function f was recovered simultaneously with the triple correlation h in Stewart/Taunsend experiments [22]. Good agreement between measured and computed values of h was achieved within the range of the reliable data [2]. In [23], isotropic homogeneous turbulence dynamics was described by a closure system of partial differential equations for the two-point double- and

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V.N. Grebenev, M. Oberlack

triple correlation functions coming from using the finite-dimensional probability density equation. The following system of equations was written:   ∂ B LL ∂ B LL 1 ∂ 4 = 4 r B LL,L + 2ν , (2.8) ∂t r ∂r ∂r a1 ∂ B LL,L ∂ B LL ∂ 1 ∂ 4 − 1/4D LL =ν r B LL,L − 3 B LL,L , ∂t ∂r ∂r r4 ∂r τ˜

(2.9)

where the first equation coincides with the von Kármán-Howarth equation, τ˜ is the quantity which characterizes the correlation time. Applying the socalled local equilibrium approximation to the second equation, the closure relationship (2.5), (2.7) can be obtained but as it was noted by Chorin [24], such approach is based only on a physical hypothesis. In the Appendix to this paper, we give a formal derivation of this formula based on studying the Riemannian invariants of characteristics of system (2.8),(2.9). This enables to find an invariant manifold admitted by (2.8),(2.9) and to construct a reduced system. Conceptually, this procedure is a similar to the approach suggested in [25]. The principle difference is that we apply the method of differential constraints [15] worked out by Cartan and Yanenko to study overdetermined systems. Finally, we note that it was in fact Hasselman [26] who was the first to hypothesize a connection between the correlation functions of the second- and third-order. His model for isotropic turbulence contains one empirical constant and a rather complicated expression for the turbulent viscosity coefficient.

3 A Model Manifold Defined by Closure of the Von Kármán–Howarth Equation First we review certain definitions and statements from Riemannian geometry. Then we construct the so-called model manifold by exploring the closure model (2.5), (2.7) for the von Kármán-Howarth equation and give a geometric interpretation of the second-order structure function D LL . To study explicitly the deformation of a family of Riemannian metrics constructed in time, we use a selfsimilar solution of the closure model for the von Kármán–Howarth equation. 3.1 Laplace-Beltrami Operator We recall the definition of some operators on a Riemannian manifold U. Consider a vector field F = Fn ∂/∂ xn on U. The operator div is determined by the formula 1  ∂ √ div F = √ ( gFn ), g n=1 ∂ xn N

Interpretation of the Second-Order Structure Function

9

 is defined where g = det gmn , and the mth component of the operator ∇ according to the formula (∇ f )m =

N 

gnm

n=1

∂f , ∂ xn

here gnm are elements of the matrix gnm −1 . Further we denote

= div ∇ the Laplace-Beltrami operator. The Laplace-Beltrami operator with a positive smooth weighted function σ (x) is defined in a similar way using the following formula  ∂ 1 √ (σ (x) gFn ). √ n σ (x) g n=1 ∂ x N

div F =

Here σ (x) presents the density of a Borel measure μ on U. If μ is the Riemannian volume, then σ (x) ≡ 1. Let Z be a Riemannian manifold which is isometric to Z  X × Y, where X is an arbitrary manifold of dim X = N1 and Y is a compact N2 dimensional manifold. Then a metric dz2 on Z is determined by dz2 = dx2 + γ 2 (x)dy2 ,

(3.10)

where γ (x) is a positive smooth function and dx2 , dy2 are metrics on X, Y correspondingly. We assume that the density σ (z) of a Borel measure μ on Z can be written as σ (z) = τ (x)η(y). Then the Laplace-Beltrami operator given on Z takes the form

Z = A + γ −2 B,

(3.11)

where A is the Laplace-Beltrami operator on X with the weighted function γ N2 τ and B denotes the Laplace-Beltrami operator defined on Y with the weighted function η [27]. As an elementary example which illustrates the above construction we consider the Laplace operator

=

∂2 ∂2 ∂2 + + ∂ x2 ∂ y2 ∂z2

written in the spherical coordinates r, ϕ, ψ (x = r sin ϕ cos ψ, y = r sin ϕ sin ψ, z = r cos ϕ)

=

1 ∂ 2∂ 1 r + 2 , r2 ∂r ∂r r2

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V.N. Grebenev, M. Oberlack

where

2 =

∂ 1 1 ∂ ∂2 sin ψ + 2 sin ψ ∂ψ ∂ψ sin ψ ∂ϕ 2

denotes the inner Laplacian on the unite sphere S2 . Then a Riemannian metric dz2 of Z = R+ × S2 is defined by the formula dz21 = dr2 + r2 dθ 2 ,

dθ 2 = dϕ 2 + sin2 ϕdψ 2 ,

(3.12)

where dθ 2 is the standard inner metric of S2 and dz21 is equivalent to the usual Euclidian metric. If we substitute a function g2 (r), g(0) = 0, g(r)  0 instead of r2 into the second term of (3.12), then the corresponding Laplace-Beltrami operator given on Z which is equipped by the metric dz22 = dr2 + g2 (r)dθ 2 takes the form

Z ,dz22 =

1 ∂ 2 ∂ 1 g (r) + 2 2 . g2 (r) ∂r ∂r g (r)

Here Z is a cylindrical domain of the radius γ = g(r) and dz22 determines another inner metric on Z . Therefore the definition of the Laplace-Beltrami operator on the cross product of Riemannian manifolds and the above examples show us that the so-called radial part A of the operator Z completely defines the form of a Riemannian metric dz given on Z  X × Y. 3.2 Metric Properties of a Model Manifold Determined by the Model Limited to Sufficiently Large Reynolds Number Let us consider a cylindrical domain Z = R+ × S2 in the correlation space K3 ( R3 ). In order to equip this cylindrical domain by an inner metric, we explore the specific form of the right-hand side of the closure model for the von Kármán–Howarth equation. So, assuming the Reynolds number to be large, the first order O(1) of Eq. 2.3 with the closure relationships (2.5), (2.7) reduces to its inviscid form 2κ2 ∂ 5 1/2 ∂ ∂ B LL = 4 r D LL B LL , ∂t r ∂r ∂r

r = |r|,

r ∈ K3 .

(3.13)

˜ LL = 2[u2 − B˜ LL ]. Then Eq. 3.13 Let q = 2r1/2 , B˜ LL (q, t) ≡ B LL (r, t) and D can be rewritten in the form ∂ B˜ LL κ2 ∂ ˜ 1/2 ∂ B˜ LL . = 9 q9 D LL ∂t q ∂q ∂q

(3.14)

Further, let Z = R+ × S2 be a manifold with the metric dz2 = dq2 + γ˜ 2 (q, tc )dθ 2

(3.15)

Interpretation of the Second-Order Structure Function

11

˜ α , α = 1/4 and β = 9/2. This manifold represents a cylindriwhere γ˜ = qβ D LL ˜ α is the radius of the hypersurface {q} × S2 cal domain such that γ˜ = qβ D LL depending on the time t. Consider the action of the operator ∂ ˜ 1/2 Z − κ2 D (3.16) LL ∂t on the longitudinal correlational function B˜ LL where Z denotes the LaplaceBeltrami operator defined on the manifold Z . Using formula (3.11), we obtain that   ∂ 1/2 ˜ − κ2 D LL Z B˜ LL (q, t) = 0, q = q(r) (3.17) ∂t due to Eq. 3.14. The direct calculations show that Eqs. 3.17 and 3.14 coincide with each other. Therefore the domain of definition of operator (3.16) evolves in time and the radius of the hypersurface {q} × S2 is determined by the formula ˜ αLL , γ˜ = qβ D

˜ LL = 2[u2 − B˜ LL ]. D

(3.18)

It means that by solutions of Eq. 3.13 we can control a deformation of the metric (3.15). Therefore if we single out a fluid volume (of spherical form) in (infinite) homogeneous isotropic flow embedded into the correlation space K3 (i.e. we introduce the correlation variables instead of physical ones), then a length scale of turbulent motion localized within this volume can be defined according to the formula (3.15) (written in the spherical coordinates r, ϕ, ψ) where γ˜ (the injectivity radius of the metric dz2 ) is determined by (3.18). We note that this spherical domain (with the punctured point r = 0) is isometric to Z with the same metric. The length scale of turbulent motion constructed by this way is a family of scales parametrized by the time t. The formula (3.18) tell us how the length scale of possible transverse displacements of fluid particles depends on the second-order structure function D LL and the correlation distance r. This kind of argument may be also used to describe the shape dynamics of this fluid volume in terms of the deformation of length scales of turbulent motion in the transverse direction. We need only to control the deformation of a measure ( length scale) of transversal displacement of fluid particles. The function γ˜ defines a measure (length scale) of these transverse displacements. The Ricci flow which shrinks homothetically a round sphere to a point serves as an illustrative example of similar phenomenon. The injectivity radius γ˜ of the metric dz2 determines the geometric structure on Z . In particular, a cylindrical domain Z is isometric to the hyperbolic space H3 (or a domain of this space) when γ˜ = sinh q. It means that the metric constructed is nonequivalent to the usual (Euclidian) metric in general. In order to study explicitly the behavior of the function γ˜ (q, t) which determines the radius of the hypersurface {q} × S2 , we use the inviscid form of the von Kármán-Howarth equation which admits the two-parameter Lie scaling subgroup and one-parameter Lie subgroup of translation transformation in

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V.N. Grebenev, M. Oberlack

time [12]. Therein this factum was applied to introduce a selfsimilar ansatz and to find a whole class of selfsimilar solutions. Other symmetries of fundamental fluid mechanics such as rotation invariance, translation invariance in time, Galilean invariance are implicitly met due to the a priori constraint of isotropic turbulence. Let us write the inviscid form of the von Kármán-Howarth equation in the following normalized form 3/2 ∂u2 (t) f (r, t) 1 ∂ = 4 r4 u2 (t) h(r, t), ∂t r ∂r

(3.19)

where f and h are respectively the normalized two-point double and triple velocity correlation. The unclosed Eq. 3.19 admits the following two scaling groups Ga1 : t∗ = t,

∗

r∗ = ea1 r,

Ga2 : t∗ = ea2 t,

f ∗ = f,

u2 = e2a1 u2 , ∗

r∗ = r,

u2 = e−2a2 u2 ,

h∗ = h,

f ∗ = f,

h∗ = h,

or in the infinitesimal form Xa1 = r

∂ ∂ , + 2u2 ∂r ∂u2

Xa2 = t

∂ ∂ . − 2u2 ∂t ∂u2

The operators Xa1 and Xa2 generate the two-parametric Lie scaling group Ga1 ,a2 : t∗ = ea2 t,

∗

r∗ = ea1 r,

u2 = e2(a1 −a2 ) u2 ,

f ∗ = f,

h∗ = h.

It is easy to check that Eq. 3.13 is invariant under the two-parametric group Ga1 ,a2 . We note that ru2 Kˆ =

3/2

(1 − f )1/2 ∂ f/∂r

t−3(σ +1)/(σ +3)

.

is a differential invariant of Ga1 ,a2 and the closure relationships suggested (2.5), (2.7) are in agreement with group properties of the inviscid form of the von Kármán-Howarth equation. Other invariants of Ga1 ,a2 are ξ=

r t2/(σ +3)

,

fˆ =

u2 f t−2(σ +1)/(σ +3)

hˆ =

,

u2

3/2

h

t−3(σ +1)/(σ +3)

,

σ=

2a2 − 3a1 . a1

where ai , i = 1, 2 are given above. Here r is scaled by the integral length scale lt ∝ t2/(σ +3) . The invariants above enable us to reduce the number of variables in Eq. 3.13 and as a result, we have the following ordinary differential equation 2κ2 d ξ 4 dξ



 d fˆ fˆ + δξ + γ fˆ = 0, dξ dξ

1/2 d

ξ (1 − fˆ) 5

(3.20)

Interpretation of the Second-Order Structure Function

13

where γ = 2(σ + 1)/(σ + 3), δ = 2/(σ + 3) and σ is undetermined at this point. Eq. 3.20 can be integrated by Loitsyansky invariant  ∞ 2  r4 f (r, t)dr,  ≡ const. (3.21) =u 0

To determine the value of the parameter σ , we rewrite (3.21) in the form (using the invariants above)  ∞  ∞ 4 −2(σ +1)/(σ +3) 10/(σ +3) 2  =u r f (r)dr = t t ξ 4 fˆ(ξ )dξ 0

0

which determines σ = 4 and hence γ = 10/7, δ = 2/7. To find a closed form solution of (3.20), we first multiply (3.20) on ξ 4 and then integrate the equation obtained by parts using the computed values α and β. As a result, we obtain the formula (see, [12])    1 + (1 − fˆ)1/2 1/2 ξ = 7κ2 ln − 2(1 − fˆ) (3.22) 1 − (1 − fˆ)1/2 which defines a solution of Eq. 3.20 in implicit form. It follows from the formula (3.22) that fˆ(ξ ) ≈ e−2ξ/3 ,

ξ  1,

fˆ(ξ ) ≈ 1 − ξ 2/3 ,

ξ  1.

The computed evolution of u2 (t) and the integral length scale lt read as follows u2 (t) ∝ (t + a)−10/7 ,

lt ∝ (t + a)2/7 ,

a ∈ R.

(3.23)

Therefore there exists a selfsimilar solution of Eq. 3.13 in the following form B LL (r, t) = u2 (t) fˆ(ξ ) ≡ (t + a)−10/7 fˆ(ξ ).

(3.24)

The exact form of the function u2 (t) makes it possible to calculate exactly the evolution of the radius γ˜ of {q} × S2 in time ˜ αLL ≡ Ar9/4 (t + a)−5/14 (1 − fˆ(rt−2/7 ))1/4 , γ˜ fˆ = qβ D

(3.25)

where A is a positive constant equals 29/2 . Remark 3.1 To the best of our knowledge, there are no published results establishing the existence and uniqueness of solutions to initial-boundary value problems for Eq. 3.13. Equation 3.13 is a nonlocal degenerate parabolic equation. This makes very delicate the proof of solvability of initial-boundary value problems for this equation. Moreover, the large time behavior of solutions and accompanying qualitative properties are not studied yet. To overcome this gap, we use the results of numerical modeling. As have been shown in [2], numerical calculations indicate some important constraints of the original closed model of the von Kármán-Howarth equation: if the Reynolds number is sufficiently large, solutions from arbitrary initial conditions relax after a small amount of time to a selfsimilar state, controlled by the large scale structure. Moreover,

14

V.N. Grebenev, M. Oberlack

the self-similar solution obtained demonstrates a good agreement between measured and analytic calculated values of fˆ and hˆ if the Reynolds number to be sufficient large. Thus we can conclude, based on numerical experiments, that the function γ˜ behaves approximately as γ˜ fˆ for large time. We give a geometric interpretation of the solution obtained (3.22). Let us rewrite (3.22) in the form   1 1 + (1 − fˆ)1/2 1/2 ˆ (3.26) ξ = −(1 − f ) + 1/2 ln 14κ2 1 − (1 − fˆ)1/2 and introduce the new variables x = ξ/14κ2 ,

y = fˆ1/2 .

(3.27)

Then (3.22) is transformed to the well-known tractrix equation [28] x = x(y) = −(a2 − y2 )1/2 +



a + (a2 − y2 )1/2 a , ln 2 a − (a2 − y2 )1/2

a=1

(3.28)

arising in differential geometry. The curve x = x(y) coincides with the element of Beltarmi surface. This is a remarkable fact since Beltrami surface is a canonical surface of the constant (sectional) negative curvature equals −1. Reflecting this surface with respect to the plane yOz of the Cartesian space R3 , we obtain the so-called pseudo-sphere: a hyperbolic manifold of the constant negative curvature. This manifold has singular points at x = 0 which forms the so-called break circle where the manifold loses smoothness. We note that according to our construction the longitudinal correlation function B LL (r, t) for each fixed time takes a constant value on the hypersurface {r} × S2 or, in another words, B LL (r, t) is a radially-symmetric function. Fixing the angle coordinate of the sphere S2 , for example ψ = ψc (or considering a cross-section of the sphere along a latitude), we can construct in the Cartesian coordinates (x, y, z) a surface of revolution generated by the curve x = x(y) (or the graphic of fˆ) which coincides with Beltrami surface. The parametric equations of the curve x = x(y) (or the graphic of fˆ) read x = ln cot

1 − cos θ, θ

y = sin θ,

0<θ <

π . 2

Revolving this curve about Ox-axis, we obtain the following family of curves xω ≡ x,

yω = sin θ cos ω,

zω = sin θ sin ω,

−∞ < ω < ∞ (3.29)

where ω is an angle of rotation of the plane XY and Eq. 3.29 present the socalled universal covering of Beltrami surface. Here the value of angle θ = π/2 corresponds the singular points of the pseudo-sphere. Parametric equations of

Interpretation of the Second-Order Structure Function

15

the surface of revolution generated by (3.24) for each fixe time tc for a crosssection of the sphere along a latitude can be written in the form sign(cos ω) 2 sin θ cos2 ω, (tc + a)10/7 π 0  ω  2π. z = sin θ sin ω, 0<θ < , 2 Equations 3.30 and 3.31 describe the evolution of this surface in time. x = ln cot

1 − cos θ, θ

y=

(3.30) (3.31)

4 Concluding Remarks The physical conclusions of this works read as: the Riemannian metric (3.15), (3.18) enables us to introduce into consideration the family of length scales (parametrized by the time t) of turbulent motion exploring the closure model for the von Kármán-Howarth equation in the limit of large Reynolds numbers; the length scales constructed by this way are compatible with the form of this closure model; the formula (3.18) tell us how the length scale of possible transverse displacements of fluid particles depends on the secondorder structure function and the correlation distance; using the selfsimilar solution (3.24) and the formula (3.25), we can estimate asymptotically decreasing the length scales of turbulent motion in the transverse direction. The frame of this consideration is limited by the case of homogeneous isotropic turbulence under the assumption of large Reynolds numbers (in the limit of large Reynolds numbers). In this paper we did not consider such questions as solvability of initialboundary value problems for Eq. 3.13, large time behavior of solutions and other accompanying topics. Acknowledgements This work was supported by DFG foundation and was partially supported by RFBR (grant No 07-01-0036). The first author grateful to Prof. V.V. Pukhnachev and G.G. Chernykh for helpful discussions. Also the authors thank Dr. M. Frewer for the useful comments and an anonymous reviewer for the reference [25].

Appendix We show that the approximation (2.5) and (2.7) can be obtained by using invariants of the characteristics of system (2.8), (2.9). So, assuming the Reynolds number to be large, the first order O(1) of Eqs. 2.8 are reduced to the system of first-order partial differential equations   ∂ B LL 1 ∂ ∂ B LL,L = 4 r4 B LL,L ≡ F 1 B LL,L , , (4.32) ∂t r ∂r ∂r   ∂ B LL,L ∂ B LL a1 ∂ B LL = 1/4D LL − 3 B LL,L ≡ F 2 B LL,L , . ∂t ∂r τ˜ ∂r

(4.33)

16

V.N. Grebenev, M. Oberlack

In order to find the Riemannian invariant I of this system we consider the operator L = Dt + λDr , where Dt and Dr are the total derivatives with respect to the variables t and r. Here λ is determined by the equation   ∂(F 1 , F 2 ) det − λE = 0, (4.34) ∂(B LL , B LL,L ) where E denotes the unit matrix. Then the Riemannian invariant I of the characteristic equation dx =λ (4.35) dt is determined by the equation L(I) = 0 due to (4.32), (4.33). Applying this definition to (4.32) and (4.33), we can rewrite this system in the form L(I)|[(4.32), (4.33)] = 0.

(4.36)

Solving Eq. 4.34, we find that this equation has the roots λ1 = 1/2D LL ,

λ2 = −1/2D LL .

In the case of λ = λ1 our equation for the invariants takes the form 1/2 Dt I(r, t, w, v) + 1/2D LL Dr I(r, t, w, v) = 0, w = B LL , (4.32), (4.33)]

v = B LL,L .

(4.37)

Equation 4.37 is a first-degree polynomial with respect to the variables v, wr and vr . As a result, we obtain the following system 4 3a1 Iw + Iv = 0, r τ˜

1/2

D LL Iw + 1/2D LL Iv = 0,

1/2

Iw + 1/2D LL Iv = 0,

1/2

It + 1/2D LL Ir = 0.

(4.38)

Suppose that D LL ≡ 0 then we find τ˜ = 3/2

a1 r 1/2

D LL

that coincides with the formula suggested in [23] which have been obtained by the dimensional analysis. Then from the Eqs. 4.38 it follows that the Riemannian invariant I1 (which corresponds to the root λ1 of Eq. 4.34) is defined by the formula  1/2 (4.39) I1 = v − 1/2D LL dw. By analogy we find

 I2 = v +

1/2

1/2D LL dw

for λ = λ2 .

(4.40)

Interpretation of the Second-Order Structure Function

17

We recall that invariants of the so-called characteristic differential Eq. (4.34) generate invariant manifolds of the corresponding system of differential equations i.e. according to [15] the following equality holds V F (Ii )|[Ii ]r = 0,

F = (F1 , F2 )

(4.41)

where V F is the vector field   ∂ ∂ ∂ Fn n + Drα (F n ) n , + ∂t n=1 ∂g ∂gα α 1 n=1 2

VF =

2

g = (g1 , g2 ) ≡ (w, v)

and [Ii ]r denotes the equation Ii = 0 and differential prolongations of this equation with respect to r. Indeed, the condition of invariance L(Ii )|(4.32), (4.33) = 0 means that Dt Ii |(4.32), (4.33) |[Ii ]r = 0, which is equivalent to (4.41). Therefore if we consider the system (4.32), (4.33) on the invariant manifold I1 = 0 then Eq. 4.32 is reduced to  ∂ B LL 1 ∂ 4 1/2 1/2D LL dw. (4.42) = 4 r ∂t r ∂r Thus we can construct a von Kármán-Howarth’s type differential model by Eq. (4.42) and using the Taylor series for the function  1/2 1/2 D LL dw. Taking into account that v(0) = 0, we write the invariant manifold in the form  1/2 |o(r)|  Cr2 . 0 = I1 = v(r) − 1/2 D1/2 dw ≡ v(r) − 1/2 r D LL wr + o(r), (4.43) Denoting appr

I1

1/2

≡ v(r) − 1/2 r D LL wr ,

as the first-order approximation of I1 , we can reduce Eq. 4.42 to κ3 ∂ ∂ B LL 1/2 ∂ B LL = 4 r5 D LL ∂t r ∂r ∂r

(4.44)

appr

on the set I1 = 0 which is similar to the von Kármán-Howarth equation but with another model constant κ3 = 0.5. Remark 4.1 Using the recommended value of the Kolmogorov constant C ≈ 1.9, and calculating 2κ2 in (2.7) which approximately equals 0.2, we find that κ3 is close to 2κ2 with respect to the order of these quantities.

18

V.N. Grebenev, M. Oberlack

References 1. Topping, P.: Lectures on the Ricci Flow. London Mathematical Society. Notes Series. Cambridge University Press, Cambridge (2006) 2. Oberlack, M., Peters, N.: Closure of the two-point correlation equation as a basis for Reynolds stress models. Appl. Sci. Res. 55, 533–538 (1993) 3. Lytkin, Yu.M., Chernykh, G.G.: On a closure of the von Kármán–Howarth equation. Dyn. Continuum Mech. 27, 124–130 (1976) 4. von Kármán, Th., Howarth, L.: On the statistical theory of isotropic turbulence. Proc. R. Soc. A 164, 192–215 (1938) 5. Lee, J.M.: Riemannian Manifolds. Graduate Text in Math. 176. Springer, New York (1997) 6. Piquet, J.: Turbulent Flows. Models and Physics. Springer, New York (1999) 7. Chernykh, G.G., Korobitsina, Z.L., Kostomakha, V.A.: Numerical simulation of isitropic dynamics. IJCFD 10, 173–182 (1998) 8. Khabirov, S.V., Ünal, G.: Group analysis of the von Kármán-Howarth equation. I : Submodels. Commun. Non. Sci. Numer. Simul. 7(1–2), 3–18 (2002) 9. Ovsyannikov, L.V.: Group Methods of Differential Equations. Academic, New York (1982) 10. Pope, S.B.: Lagrangian PDF methods for turbulent flows. Annu. Rev. Fluid Mech. 26, 23–63 (1994) 11. Pope, S.B.: On the relationship between stochastic Lagrangian models of turbulence and secon-moment closure. Phys. Fluids 6, 973–985 (1994) 12. Oberlack, M.: On the decay exponent of isotropic turbulence. In: Proceedings in Applied Mathematics and Mechanics, 1. http://www.wiley-vch.de/publish/en/journals/ alphabeticIndex/2130/?sID= (2000) 13. Monin, A.S., Yaglom, A.M.: Statistical Hydromechanics. Gidrometeoizdat, St.-Petersburg (1994) 14. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1986) 15. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: The Differential Constraints Method and its Application to Gas Dynamics. Nauka, Novosibirsk (1984) 16. Oberlack, M., Busse, F.H. (eds.): Theories of Turbulence. Lecture Notes in Computer Science. 442. Springer, New York (2002) 17. Robertson, H.P.: The invariance theory of isotropic turbulence. Proc. Camb. Philol. Soc. 36, 209–223 (1940) 18. Arad, I., L’vov, S.L., Procaccia, I.: Correlation functions in isotropic turbullence: the role of symmetry group. Phys. Rev. E. 59, 6753–6765 (1999) 19. Batchelor, G.K., Townsend, A.A.: Decay of turbulence in the final period. Proc. R. Soc. A 194, 527–543 (1948) 20. Millionshtchikov, M.: Isotropic turbulence in the field of turbulente viscosity. Lett. JETP. 8, 406–411 (1969) 21. Grebenev, V.N., Oberlack, M.: A Chorin-type formula for solutions to a closure model of the Kármán–Howarth equation. J. Nonlinear Math. Phys. 12, 1–9 (2005) 22. Steward, R.W., Townsend, A.A.: Similarity and self-preservation in isitropic turbulence. Philos. Trans. R. Soc. A 243(827), 359–386 (1951) 23. Onufriev, A.: On a model equation for probability density in semi-empirical turbulence transfer theory. In: The Notes on Turbulence. Nauka, Moscow (1994) 24. Chorin, A.: Lecture Notices in Math, 615. Springer, Berlin (1977) 25. Chen, G.Q., Levermpre, C.D., Liu, T.-P.: Hyperboloic conservation kaws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47(6), 787–830 (1994) 26. Hasselmann, K.: Zur Deutung der dreifachen Geschwindigkeits korrelationen der isotropen Turbulenz. Dtsh. Hydrogr. Zs. 11, 207–211 (1958) 27. Svetlov, A.V.: A discreteness criterion for the spectrum of the Laplace–Beltrami operator on quasimodel manifolds. Siberian Math. J. 43, 1103–1111 (2002) 28. Michenko, A.S., Fomenko, A.T.: Lectures on Differential Geometry and Topology. Factorial, Moscow (2000)

Math Phys Anal Geom (2009) 12:19–45 DOI 10.1007/s11040-008-9050-y

Heisenberg-Integrable Spin Systems Robin Steinigeweg · Heinz-Jürgen Schmidt

Received: 20 June 2007 / Accepted: 24 October 2008 / Published online: 21 November 2008 © Springer Science + Business Media B.V. 2008

Abstract We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P. For these systems the time evolution can be explicitly calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four as vertex-induced sub-graphs. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed. Keywords Completely integrable systems · Heisenberg spin systems Mathematics Subject Classifications (2000) 70 H 06 · 37 J 35 · 81 Q 05 · 94 C 15 · 82 D 40

R. Steinigeweg · H.-J. Schmidt (B) Fachbereich Physik, Universität Osnabrück, Barbarastr. 7, 49069 Osnabrück, Germany e-mail: [email protected]

20

R. Steinigeweg, H.-J. Schmidt

1 Introduction Integrable physical systems are rare; nevertheless there exists a vast literature about these systems, let it be classical or quantum ones. They are interesting in their own right but also in connection with properties like chaotic behavior [1, 2], level statistics [3, 4] or transport properties [5–7]. In this paper we deal with a special class of integrable spin systems, called Heisenberg-integrable systems. This is motivated by investigations on magnetic molecules. The synthesis of these molecules has undergone rapid progress in recent years building on successes in coordination and polyoxometalate chemistry [8–11]. Each of the identical molecular units can contain as few as two and up to several dozen paramagnetic ions. It appears that in the majority of these molecules the localized single-particle magnetic moments (“spins”) couple antiferromagnetically and the spectrum is rather well described by the Heisenberg model [12]. The synthesis of these magnetic molecules has additionally aroused the interest in small spin systems, as opposed to the traditional topic of infinitely large spin lattices. Nevertheless, small spin systems may have large Hilbert spaces: Since the dimension of the Hilbert space for N spins of spin quantum number s, given by (2s + 1) N , grows rapidly with N and s, the numerical evaluation of all energy eigenvalues may be practically impossible. Analogous remarks apply to the study of classical spin systems, where s → ∞. Hence it is useful to know as much as possible about integrable spin systems where the Hamiltonian can be analytically diagonalized. Integrable spin systems may additionally be used to test numerical procedures or even to construct numerical integrators [13]. Here we want to systematize the scattered knowledge on integrable spin systems of a particular kind, the Heisenbergintegrable systems. Classical spin systems are examples of Hamiltonian mechanical systems. Hence the term “integrable” has a precise meaning in the context of the Liouville-Arnold theorem [14]. It requires that there exist N independent, commuting constants of motion, where N denotes the number of spins and “commutation” is understood w. r. t. the Poisson bracket. For integrable systems one can find so-called action-angle variables In , ϕn such that ∂H I˙n = − = 0, ∂ϕn

ϕ˙n =

∂H = ωn = const. , n = 1, . . . , N . ∂ In

(1)

Hence the equations of motion can be solved explicitly if the integrations involved in the definitions of the In , ϕn can be performed, see [14]. The task to independently characterize the class of all integrable spin systems IS seems extremely difficult. Most published work on integrable spin systems deals with special examples and numerical case studies [1, 2]. Also in this article we will not characterize IS itself, but certain subclasses of IS . For quantum systems the corresponding notion of “integrability” is less precise. Although some prominent examples of integrable classical systems

Integrable Spin Systems

21

have solvable quantum mechanical counterparts, including the harmonic oscillator, the Kepler problem and the two center Kepler problem, there is no comparable general theory of integrable quantum systems. For example, the general Heisenberg spin triangle with different coupling constants is an integrable classical system, but we do not know of any procedure to analytically calculating the eigenvectors and eigenvalues of the general quantum spin triangle for arbitrary individual spin quantum number s. Moreover, some quantum systems like the s = 12 Heisenberg spin chain of length N are considered as completely integrable, for example by quantum inverse scattering methods [15], but possess non-integrable classical counter-parts for N ≥ 5. However, for the subclasses of integrable spin systems to be considered below the eigenvalue problem of the corresponding quantum spin Hamiltonian can be analytically solved. In this article we investigate a subclass HIS ⊂ IS of the class of integrable spin systems, called Heisenberg integrable systems, or, shortly, H-integrable systems. They are defined by the extra condition that N − 1 of the N constants of motion, as well as the Hamiltonian H itself, are of Heisenberg type, i. e. consist of linear combinations of scalar products of spin vectors: E(n) =

 μ<ν

(n) sμ · sν , n = 1, . . . , N − 1. Eμν

(2)

The remaining N-th constant of motion is chosen as the 3−component of the total spin S(3) . We conjecture that HIS = IS if H is of Heisenberg type, but we have not proven this although we have some evidence from numerical studies of Ljapunov exponents for small spin systems, see also [16]. We did not obtain an independent characterization of HIS itself, but only for two subclasses HIG ⊂ BS ⊂ HIS called “Heisenberg integrable spin graphs” and “B -partitioned spin systems”. “Spin graphs” are systems with Heisenberg Hamiltonians  H= Jμν sμ · sν , (3) μ<ν

satisfying Jμν ∈ {0, 1}. Obviously, the coupling scheme of such a system can be represented by an undirected graph, the N vertices v ∈ V of which correspond to the N spins and the edges (μ, ν) ∈ E to those pairs of spins where Jμν = 1. For our purposes it is convenient to consider a special notion of a “subsystem”, see Section 2. In the case of a spin graph G = (V , E ) a subsystem will consist of a subset V  ⊂ V of vertices together with all edges e ∈ E which connect vertices of V  . In other words, the sub-graph G  = (V  , E  ) is obtained by removing a number of vertices along with any edges which contain a removed vertex. This notion of a subsystem is equivalent to what graph theorists call a “vertex-induced subgraph”, see [25].

22

R. Steinigeweg, H.-J. Schmidt

All spin graphs with N ≤ 4 turn out to be H-integrable, with the exception of the 4-chain. One main result of this article is that a spin graph is H-integrable iff it contains no 4-chain as a subsystem (in the sense explained above) iff it is the union of two uniformly coupled or disjoint H-integrable subsystems (“uniform or disjoint union”). By recursively applying the uniform union property we obtain a partition of the whole spin graph into smaller and smaller H-integrable subsystems with uniform or vanishing coupling. This sequence of partitions can be encoded in a binary “partition tree” B . Removing the condition Jμν ∈ {0, 1} we arrive at the slightly more general notion of B -partitioned spin systems for which the time evolution can be analytically calculated. The observation that the uniform union of two integrable systems is again integrable is certainly not new, see e. g. [17, 18] for a special case concerning quantum spin systems or [19] for classical systems. However, the use of partition trees in order to obtain the time evolution or the eigenvectors in closed form seems to be novel. Moreover, our B -partitioned spin systems are closely related to integrable Hamiltonians constructed from co-algebras, see [20], Theorem 2. More precisely, if we specialize B to be a monotone partition (for the definition see Section 5.1) then our N − 1 quadratic constants of motion can be expressed by the Casimir elements C(m) , 2 ≤ m ≤ N defined in [20] in the case of the Lie algebra g = so(3). Recall that the construction in [20] includes the GaudinCalogero systems [21, 22] for the choice of g = so(2, 1). It seems that the results of [20] also hold for the case of a general partition tree B as considered in this paper. Our article is organized as follows. In Section 2 we present the pertinent definitions and first results on classical integrable or H-integrable spin systems. Further results on subsystems and uniform unions are contained in Section 3. Among these is Theorem 1 saying that any subsystem of an H-integrable spin system is again H-integrable. Section 4 contains our results on spin graphs. For example, Theorem 2 states that each H-integrable spin graph is the uniform union of two H-integrable subsystems. Finally we will prove that a spin graph is H-integrable iff it does not contain any 4-chain, see Theorem 3. As an application, we enumerate all connected spin graphs up to N = 5 in the Appendix. If they are H-integrable we indicate the uniform decomposition as well as the N commuting constants of motion; if they are not we display some 4-sub-chain. The next Section 5 is devoted to the explicit form of the time evolution for B -partitioned spin systems, see Theorem 4. It turns out that their time evolution can be described by a suitable sequence of rotations about constant axes. This is closely related to the definition of action-angle variables satisfying (1), as we will show in Section 5.2. In Section 5.3 we will sketch how to calculate the eigenvectors and eigenvalues of the Hamiltonian in the quantum version of B -partitioned spin systems. Section 6 contains a summary and an outlook.

Integrable Spin Systems

23

2 Definitions and First Results Classical spin configurations are most conveniently represented by N-tuples of unit vectors s = (s1 , . . . , s N ), |sμ |2 = 1 for μ = 1, . . . , N. The compact manifold of all such configurations is the phase space of the spin system    P = P N = (s1 , . . . , s N ) |sμ |2 = 1 for μ = 1, . . . , N .

(4)

The three components siμ (i = 1, 2, 3) of the μ-th spin vector can be viewed as functions on P siμ : P −→ R .

(5)

In order to formulate Hamilton’s equation of motion we need the Poisson bracket between two arbitrary smooth functions f, g : P −→ R .

(6)

The Poisson bracket has to satisfy a couple of general properties, namely bilinearity, antisymmetry, Jacobi identity and Leibniz’ rule, see e. g. [23] 10.1. Hence it suffices to define the Poisson bracket between functions of the form (5): 

3   siμ , sνj ≡ δμν ijk skμ ,

(7)

k=1

where δμν denotes the Kronecker symbol and ijk the components of the totally antisymmetric Levi-Civita tensor. This definition turns P into a Poisson manifold. We will sketch a more abstract way to endow P N with a Hamiltonian structure: Let G be a Lie group and g∗ the dual of its Lie algebra. g∗ is endowed with a canonical Poisson bracket and, moreover, is a disjoint union of the orbits of the co-adjoint action of G upon g∗ . Every such orbit is a natural symplectic manifold, see [23], Chapter 14. The phase space P N of a classical spin system results if G is taken as the N-fold direct product of the rotation group SO(3). In this case g∗ ∼ = (R3 ) N can be given the structure of a product of Euclidean spaces, unique up to an arbitrary positive factor, such that G operates isometrically on g∗ . Then P N is the co-adjoint orbit consisting only of unit vector configurations. Hence the ijk in (7) has its origin in the Lie bracket of the Lie algebra of SO(3), which is also the origin of the commutation relations of angular momenta in quantum mechanics. In the sequel we will, however, not make use of this abstract approach. Having defined the Poisson bracket, we can write down the differential equation corresponding to a given smooth function H : P −→ R:   d i s = siμ , H , μ = 1, . . . , N, i = 1, 2, 3 . dt μ

(8)

24

R. Steinigeweg, H.-J. Schmidt

The r. h. s. of (8) can be viewed as the vector field X H on P generated by the function H. If H is the Hamiltonian of the spin system, (8) is Hamilton’s equation of motion and X H is the corresponding Hamiltonian vector field. It is complete since P is compact, see [24] Cor. 4.1.20. Generally, we define Ft (H)s(0) = s(t), t ∈ R ,

(9)

where s(t) = (s1 (t), . . . , s N (t)) is the solution of (8) with initial value s(0). Ft (H) : P −→ P is called the flow of H and is defined for all t ∈ R due to the completeness of the Hamiltonian vector field. Lemma 1 Let H, K : P −→ R be smooth functions. Then the flows Ft (H) and Ft (K) commute iff {H, K} = 0. Proof The if-part is a standard result, since the commutation of the flows is equivalent to 0 = [X H , X K ], see [24] 4.2.27, and [X H , X K ] = −X{H,K} , see [23] 5.5.4. For the only-if-part we conclude 0 = [X H , X K ] = −X{H,K} , hence {H, K} = c = const.. For c = 0 the differential equation d K(s(t)) = {K, H} = −c dt

(10)

has unbounded solutions, which is impossible for compact P . Hence {H, K} = 0.   For the rest of this section we consider a fixed Heisenberg Hamiltonian H : P −→ R. It will be convenient to identify the spin system with its Hamiltonian. A constant of motion is a smooth function f : P −→ R which commutes with the Hamiltonian: { f, H} = 0. H is said to be of Heisenberg type, or, short, a Heisenberg Hamiltonian, if it is of the form H(s) =



Jμν sμ · sν .

(11)

μ<ν

The real numbers Jμν are called coupling constants. It will be convenient to set Jνμ = Jμν for μ < ν. Define the total spin vector S ≡

N 

sμ

(12)

μ=1

 A Heisenberg with components S(i) , i = 1, 2, 3 and square S2 ≡ S · S. Hamiltonian commutes with all components of the total spin and its square:     0 = H, S2 = H, S(i) , i = 1, 2, 3 .

(13)

Integrable Spin Systems

25

Let A = ∅ be any subset of {1, . . . , N} and a = |A|. Then P A denotes the phase space of the subsystem A, i. e. the manifold of all spin configurations of the form s A = (sμ1 , . . . , sμa ) such that μ1 < μ2 < . . . < μa and |sμi |2 = 1 for all μi ∈ A. The Hamiltonian H A of the subsystem A will be defined by 

H A (s A ) =

Jμν sμ · sν .

(14)

μ<ν μ,ν∈A

 Similarly, we define S A = μ∈A sμ together with its components S(i) A and its square S2A . Next we consider a decomposition of {1, . . . , N} into two disjoint subsets, ˙ such that A, B = ∅. Let further N A ≡ |A| and N B ≡ |B|. {1, . . . , N} = A∪B The Heisenberg Hamiltonian H is accordingly decomposed into three parts: H(s) =

 μ<ν μ,ν∈A

Jμν sμ · sν +

 μ<ν μ,ν∈B

Jμν sμ · sν +



Jμν sμ · sν

(15)

μ∈A,ν∈B

≡ H A + H B + H AB .

(16)

Here and in the sequel we identify functions of Heisenberg type defined on the phase space of a subsystem P A with their unique extension to the total phase space P , defined by Jμν = 0 for all μ, ν with μ ∈ / A or ν ∈ / A. If the coupling constants Jμν , μ ∈ A, ν ∈ B occurring in H AB have all the same non-zero value, say, Jμν = c AB ∈ R, c AB = 0, we will call the system H the uniform union of the subsystems H A and H B . If the analogous condition holds with c AB = 0 we call the system H the disjoint union of the subsystems H A and H B . A Heisenberg system is called connected if it is not the disjoint union of two subsystems. A Heisenberg system where all coupling constants are non-zero and have the same value will be called a pantahedron. Definition 1 A Heisenberg spin system H is called Heisenberg integrable, or, short, H-integrable, if there exist N − 1 independent constants of motion E(n) of Heisenberg type which commute pairwise: 

 E(n) , E(m) = 0 for all n, m = 1, . . . , N − 1 .

(17)

The Heisenberg Hamiltonian H, which commutes with all E(n) , will in general be a linear combination of the E(n) and hence need not be explicitly included in the set of the independent constants of motion.

26

R. Steinigeweg, H.-J. Schmidt

The above condition of independence means that there exists some s ∈ P such that the set of covectors {dE(1) (s), . . . , dE(N−1) (s)} is linearly independent. It follows that this condition is also satisfied in some neighborhood of s ∈ P . But it cannot hold globally: If you take some F in the linear span of the E(n) such that s ∈ P is a critical point of F, i. e. dF(s) = 0, then it is obviously violated. Later we will derive a simple criterion for the independence of a number of Heisenberg constants of motion, see Proposition 1. If a connected spin system is H-integrable, it can be easily shown that {E(1) , . . . , E(N−1) , E(N) = S(3) } will be a set of N independent, commuting constants of motion. Hence any H-integrable system is also integrable in the sense of the Liouville-Arnold theorem. We conjecture that the converse is also true. For an integrable spin system the N constants of motion E(n) are not uniquely determined. First, one can consider linear transformations F (n) =

N 

Anm E(m) , n = 1, . . . , N ,

(18)

m=1

where the Anm are the entries of an invertible matrix. These transformations leave invariant the space E of functions spanned by the E(n) n = 1, . . . , N. But also the space E need not be uniquely determined by the Hamiltonian H: Consider the disjoint union H of two H-integrable subsystems H A and H B . Then one could either consider the union of the two sets of independent, (3) commuting constants of motion for the subsystems, including S(3) A and S B , or, alternatively, the union of the Heisenberg constants of motion of the subsystems together with S2 − S2A − S2B and S(3) . Since the second choice of the E(n) is always possible, our Definition 1 entails that the disjoint union of H-integrable spin systems will be again H-integrable, see Proposition 3. For later reference we mention the following well-known fact without proof: Lemma 2 For any spin system with N spins there exist at most N independent, commuting constants of motion. A spin graph is a Heisenberg spin system where all non-zero coupling constants have the same strength J = 0. Without loss of generality we may assume Jμν ∈ {0, 1}. As explained in the introduction, the system can be represented as an undirected graph with N vertices. Of course, the above definition of a connected Heisenberg spin system coincides with the graph-theoretic notion of connectedness if the system is a spin graph. Recall that subsystems of spin graphs are understood as vertex-induced subgraphs, see above. The following fact is well-known, see [25], Theorem 1.6: Lemma 3 In a connected spin graph with N ≥ 2 vertices one can remove two suitable vertices such that the remaining subsystem is still connected.

Integrable Spin Systems

27

In order to evaluate the Poisson bracket between functions of Heisenberg type and to argue with the resulting equations we need the following lemmas which are easily proven: Lemma 4     (i) sμ · sν , sλ · sν = sμ · (sλ × sν ) = det sμ , sλ , sν 3   Let vectors, then  k, ∈ R be constant     sν (ii) sμ · sν , k · sν = det sμ , k,      ,  sν (iii) k · sν ,  · sν = det k, Lemma 5 If one of the following equations N 

 μ · sμ = 0, C

(19)

Cμ ν sμ · sν = 0,

(20)

   μ ν · sμ × sν = 0, C

(21)

μ=1 N  μ<ν N  μ<ν N 

  Cμ ν λ det sμ , sν , sλ = 0

(22)

μ<ν <λ

holds for all (s1 , . . . , s N ) ∈ P then all coefficients of the corresponding equation must vanish. Proof By induction over N using the replacement s N+1 → −s N+1 .

 

Now we can formulate a criterion for the commutation of two functions of Heisenberg type: Lemma 6 Let H be a Heisenberg system and E a function of Heisenberg type. E commutes with H iff Eμ ν (Jμ λ − Jν λ ) + Eμ λ (Jν λ − Jμ ν ) + Eν λ (Jμ ν − Jμ λ ) = 0

(23)

for all μ < ν < λ ≤ N. Proof The lemma is proven in a straightforward manner by using Lemma 4(i), cyclic permutations of triple products and (22).  

28

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Next we will show that the independence of M functions of Heisenberg type is equivalent to the linear independence of the corresponding symmetric matrices. Proposition 1 Let E(n) : P −→ R be M functions of Heisenberg type, i. e. E(n) (s) =

 μ<ν

(n) sμ · sν , n = 1, . . . , M . Eμν

(24)

(n) and denote by E(n) the symmetric N × N-matrix with entries E(n) μν = Eμν and (n) Eμμ = 0 for μ, ν = 1, . . . , N − 1. Then the following conditions are equivalent:

(i) There exists an s ∈ P such that the set of covectors {dE(n) (s)|n = 1, . . . , M} is linearly independent. (ii) The set of matrices {E(n) |n = 1, . . . , M} is linearly independent. Proof We will prove the equivalence of the negations of (i) and (ii). (i) ⇒ (ii). Assume that {E(n) |n = 1, . . . , M} is linearly dependent. That is,  (n) there exists a non-vanishing real vector (λ , . . . , λ ) such that λ E = 0. 1 M n n  (n) λ E (s) = 0 for all s ∈ P and hence 0= It follows that E(s) ≡ n n  (n) (n) (n) d λ E (s) = λ dE (s). Hence the set of covectors {dE (s)|n = n n n n 1, . . . , M} is linearly dependent for all s ∈ P . (ii) ⇒ (i). It is possible to invert the sequence of arguments of the first part ? E = 0. Here we can only conclude of the proof, except at the step dE = 0 ⇒ E = c = const. and obtain the apparently weaker condition c = E(s) =

 n

λn E(n) (s) =



(n) sμ · sν λn Eμν

(25)

n,μ<ν

 sν by −sν for fixed ν yields μ Eμν sμ · sν = 0. By sumfor all s ∈ P . Replacing ming over ν we obtain μ<ν Eμν sμ · sν = 0 and, by (20), Eμν = 0 for all μ < ν.  Hence n λn E(n) = 0 and the set {E(n) |n = 1, . . . , M} is linearly dependent.     We note that (23) can be viewed as a system of N3 linear equations for   the N2 unknowns Eμν . An H-integrable system admits at least N − 1 linearly independent solutions, according to Proposition 1. In general, it will admit more solutions, but only N − 1 of these will lead to commuting constants of motion. Hence, in the H-integrable   case, the matrix M of the system of linear equations (23) has the rank r ≤ N2 − (N − 1). For N = 4 the rank condition   r = N2 − (N − 1) = 3 is even sufficient since it implies the existence of a

Integrable Spin Systems

29

constant of motion E which is not of the form λH + μS2 . After some algebra we obtain: Proposition 2 A Heisenberg system with N   J13 − J23 J23 − J12   J14 − J24 0  0 J14 − J34

= 4 spins is H-integrable iff   0  J24 − J12  = 0 . J34 − J13 

(26)

This criterion can be used to independently check the results on spin graphs with N = 4, see Appendix.

3 Subsystems and Uniform Union In this section we will collect some general results on H-integrable systems in connection with subsystems and uniform unions. Lemma 7 Let E be a Heisenberg constant of motion of a Heisenberg system H and ∅ = A ⊂ {1, . . . , N}. Then the restriction E A is a Heisenberg constant of motion of the subsystem H A . Proof The lemma follows from Lemma 6 since the restricted functions E A and H A commute iff the equations (23) hold for μ < ν < λ with μ, ν, λ ∈ A.   Proposition 3 If a Heisenberg system H is the uniform or disjoint union of two H-integrable subsystems H A and H B , then H itself is H-integrable. Corollary 1 Each pantahedron is H-integrable. Proof Let E A be one of the N A − 1 independent, commuting Heisenberg constants of motion of H A . In particular, E A commutes with H A and also with H B since B ∩ A = ∅. Since E A is a function of Heisenberg type it commutes with S2 , S2A and S2B , hence also with H AB since H AB =

  1 c AB S2 − S2A − S2B . 2

(27)

It follows that E A commutes with H. The same holds for a corresponding constant of motion E B of the second subsystem. Hence the N A − 1 functions E A together with the N B − 1 functions E B and S2 − S2A − S2B form a set of N − 1 independent, commuting constants of motion of Heisenberg type. This means that H is H-integrable.   The converse of Proposition 3 is not true: The general spin triangle is a Heisenberg spin system with N = 3 and three different coupling constants. It

30

R. Steinigeweg, H.-J. Schmidt

has H and S2 as independent, commuting constants of motion and is hence H-integrable, but it is not the uniform or disjoint union of two H-integrable subsystems. Next we will show that H-integrability is heritable to subsystems. Theorem 1 Any subsystem H A of an H-integrable system is itself H-integrable. Proof Consider N − 1 independent, commuting constants of motion E1 , . . . , E N−1 of the form Ei =



Ei, μ ν sμ · sν .

(28)

μ<ν

These constants of motions span a linear space F . According to the assumptions of the theorem {1, . . . , N} is the disjoint union of two nonempty subsets A and B such that N = N A + N B with N A = |A| and N = |B|. We arrange the coefficients Ei, μ ν in the form of a matrix E with    NB rows and N − 1 columns. The first N2A rows contain the coefficients with 2   μ < ν and μ, ν ∈ A; the next N2B rows contain the coefficients with μ < ν and μ, ν ∈ B and finally the remaining N A N B rows those with μ ∈ A, ν ∈ B. In this way the matrix is divided into three blocks, see the following figure.

E1 E2

...

EN–1

(N2 ) A

...

(N2)

(N2 ) B

... ...

NANB

N–1 Next this matrix will be transformed into a lower triangular form by elementary Gauss transformations. We allow arbitrary permutations of columns

Integrable Spin Systems

31

and arbitrary permutations of rows within the three blocks, see the following figure.

F1 F2

...

F N–1

(N2 ) A

0

(N2)

(N2 ) B

NANB dA

dB N–1

dAB

The resulting matrix F begins with d A linearly independent columns spanning a linear space F A . d A is the maximal number of independent constants of motions of the subsystem A obtained as restrictions to A of functions from F . The next d B columns of F span the linear space F B . d B is the maximal number of independent constants of motions of the subsystem B obtained as restrictions to B of functions from F which vanish on A. The remaining d AB columns span the linear space F AB of functions from F vanishing on A and on B. Since elementary Gauss transformations do not change the rank of the matrix, we have d A + d B + d AB = N − 1 .

(29)

For sake of simplicity we identify the columns of F with the corresponding constants of motion. By Lemma 7, the restrictions to A of F1 , . . . , Fd A will be constants of motion of H A . According to Lemma 2 there are at most N A − 1 independent constants of motion. The analogous argument holds for Fd A +1 , . . . , Fd A +d B and the subsystem H B . Hence d A ≤ N A − 1 and d B ≤ N B − 1 .

(30)

It follows that N − 1 = d A + d B + d AB ≤ N A − 1 + N B − 1 + d AB = N − 2 + d AB , hence d AB ≥ 1 .

(31)

32

R. Steinigeweg, H.-J. Schmidt

Next we want to show that d AB ≤ 1. In this case we are done: d A < N A − 1 would imply N − 1 = d A + d B + d AB < N A − 1 + N B − 1 + 1 = N − 1 which is a contradiction. Hence d A = N A − 1 and the subsystem H A would be H-integrable. Proving d AB ≤ 1 is equivalent to show that F AB is at most one-dimensional, i.e. that the ratios of the coefficients Fμλ /Fνκ with μ, ν ∈ A and λ, κ ∈ B are uniquely determined. Hence consider some F ∈ {Fd A +d B +1 , . . . , F N−1 }. Its restrictions are F A = F B = 0. For μ, ν ∈ A and λ ∈ B Lemma 6 yields Fμ λ (Jν λ − Jμ ν ) + Fν λ (Jμ ν − Jμ λ ) = 0.

(32)

Then the following ratio of coefficients Fμ λ Jμ ν − Jμ λ = Fν λ Jν λ − Jμ ν

(33)

is uniquely determined, except in the case where the nominator Jμ ν − Jμ λ and the denominator Jν λ − Jμ ν vanish simultaneously: Jμ ν = Jμ λ = Jν λ .

(34)

Define, for fixed μ ∈ A and λ ∈ B, M(μ, λ) to be the set of ν ∈ A satisfying (34). If M(μ, λ) = A then H A is a pantahedron and hence H-integrable. If M(μ, λ) = A then there exists a κ ∈ A such that κ ∈ / M(μ, λ) and the ratios Fμλ / Fκλ and Fνλ / Fκλ are uniquely determined. Hence also the ratio Fμ λ Fμλ Fκλ = · Fν λ Fκλ Fνλ

(35)

will be uniquely determined. By analogous reasoning, also the ratio Fνλ / Fνκ with ν ∈ A and λ, κ ∈ B, and hence Fμλ /Fνκ with μ, ν ∈ A and λ, κ ∈ B will be uniquely determined. This completes the proof.   The preceding proof also shows: Corollary 2 Let H be an H-integrable spin system with two subsystems H A and ˙ H B such that {1, . . . , N} = A∪B. According to Theorem 1, H A and H B are also H-integrable. Then the N − 1 independent, commuting constants of motion F1 , . . . , F N−1 can be chosen such that F1 , . . . , F N A −1 are independent, commuting constants of motion of H A and vanish on B, (ii) F N A , . . . , F N−2 are independent, commuting constants of motion of H B and vanish on A, (iii) F N−1 vanishes on A and on B. (i)

Integrable Spin Systems

33

Proof It remains to show that the F1 , . . . , F N A −1 of (i) vanish on B. This can be done by further Gauss transformations which add suitable multiples of columns of (ii) to the columns of (i).  

4 Spin Graphs As explained in Section 2, spin graphs are Heisenberg spin systems such that the coupling constants satisfy Jμ ν ∈ {0, 1}. For these systems, H-integrability can be completely analyzed. According to Proposition 2 the uniform or disjoint union of H-integrable systems is again H-integrable, but not all H-integrable systems are obtained in this way. However, all H-integrable spin graphs are the uniform or disjoint unions of H-integrable subsystems, as we will show. This means that there is a construction procedure by which one can compose all H-integrable spin graphs from small constituents. Starting with two single spins, which are trivially H-integrable, we can either form a disjoint union or a spin dimer. The uniform union of a dimer and a single spin yields a uniform triangle; the uniform union of a single spin with a pair of disjoint spins is a 3-chain, etc. Remarkably, the 4-chain cannot be obtained in this way and is hence not H-integrable. Our first result is: Lemma 8 Each connected H-integrable spin graph with N > 1 vertices is the uniform union of two subsystems. Proof The proof will be performed by induction over N. For N = 2 the theorem holds since the dimer is the uniform union of two single spins. Next we assume the theorem to hold for all spin graphs with N or less vertices and consider a connected spin graph with N + 1 vertices and Hamiltonian H. According to Lemma 3 we may assume that the subsystem H N with vertices {1, . . . , N} is connected and, by Theorem 1, H-integrable. We denote by H N+1 the single spin system with vertex N + 1. By the induction hypothesis and since N > 1, H N is the uniform union of two H-integrable subsystems H A and H B , ˙ and A, B = ∅. where {1, . . . , N} = A∪B H A is further decomposed into subsystems H 0A and H 1A with vertex sets A0 and A1 , respectively. A0 consists of those spins in A which do not couple to N + 1; A1 consists of the remaining spins in A which thus uniformly couple to N + 1. The analogous decomposition is performed w. r. t. H B . Since A, B = ∅ we must have A0 = ∅ or A1 = ∅, and, similarly, B0 = ∅ or B1 = ∅. In the case A0 = B0 = ∅ the proof is done since this means that H N couples uniformly with H N+1 . The case A1 = B1 = ∅ can be excluded since it implies that H is disconnected. Hence it suffices to consider the case A1 = ∅ and B0 = ∅ in what follows. The other remaining case A0 = ∅ and B1 = ∅ can

34

R. Steinigeweg, H.-J. Schmidt

be treated completely analogously. The situation is illustrated in the following figure. HA

HN+1

1 HA

0 HA

1 HB

0 HB HB

If A0 = ∅, the total system H is a uniform union of the two subsystems with vertex sets A and {N + 1} ∪ B and the proof is done. Hence we may assume A0 = ∅. If the coupling between the subsystems H 0A and H 1A is uniform we may rearrange the decomposition by setting A0 = ∅ and B0 = A0 ∪ B0 , leaving A1 and B1 unchanged. But this case has already be considered above. Hence we may assume that the coupling between H 0A and H 1A is non-uniform. By Corollary 2 (iii) and since H N is H-integrable, the total system H possesses a non-zero constant of motion of the form  Eμ N+1 sμ · s N+1 . (36) E= μ < N+1

Lemma 6 implies Eμ N+1 (Jμ λ − Jλ N+1 ) + Eλ N+1 (Jμ N+1 − Jμ λ ) = 0,

(37)

for all μ < λ < N + 1 since Eμ λ = 0. We will show that the case A0 , A1 , B0 = ∅ is in contradiction to the above-mentioned fact that E is non-zero. To this end we consider (37) in the following four cases: –

μ0 ∈ B0 and λ1 ∈ A1 (Jμ0 λ1 = Jλ1 N+1 = 1 and Jμ0 N+1 = 0) ⇒ Eλ1 N+1 = 0 .

–

μ1 ∈ B1 and λ0 ∈ A0 (Jμ1 λ0 = Jμ1 N+1 = 1 and Jλ0 N+1 = 0) ⇒ Eμ1 N+1 = 0 .

–

(39)

μ0 ∈ B0 and λ0 ∈ A0 (Jμ0 N+1 = Jλ0 N+1 = 0 and Jμ0 λ0 = 1) ⇒ Eμ0 N+1 = Eλ0 N+1

–

(38)

(40)

λ0 ∈ A0 , λ1 ∈ A1 and Jλ0 λ1 = 0 (Jλ0 N+1 = 0 and Jλ1 N+1 = 1) ⇒ Eλ0 N+1 = 0 .

(41)

Integrable Spin Systems

35

Since the coupling between H 0A and H 1A is non-uniform, there exist λ0 ∈ A0 and λ1 ∈ A1 such that Jλ0 λ1 = 0. For this λ0 the coefficient Eλ0 N+1 vanishes by (41). Equation (40) then yields Eμ0 N+1 = 0 for all μ0 ∈ B0 and, further, Eλ0 N+1 = 0 for all λ0 ∈ A0 . By the equations (38) and (39) the remaining coefficients of E vanish, which leads to a contradiction.   Lemma 8, Theorem 1 and Proposition 3 together yield: Theorem 2 Each H-integrable spin graph is the uniform or disjoint union of two H-integrable subgraphs. It follows from Theorem 2 that all spin graphs with N ≤ 3 are H-integrable, but that the chain with N = 4 is not H-integrable since it is not the uniform union of smaller systems. By virtue of Theorem 1 every spin graph containing a 4-chain will not be H-integrable. The converse is also true and yields the following graph-theoretic characterization of H-integrable spin graphs. Theorem 3 A spin graph is not H-integrable iff it contains a chain of length four as a vertex-induced sub-graph. Proof It remains to show the only-if-part. This will be done by induction over N. For N = 4 the theorem is proven by a complete classification of all connected spin graphs, see the Appendix. Next we assume that the theorem holds for all spin graphs with N or less spins and consider a spin graph H with N + 1 vertices which is not H-integrable. If H is the union of two disjoint subsystems, at least one of them is not H-integrable by Proposition 3 and hence contains a 4-chain by the induction assumption. Thus we may assume that H is connected. By virtue of Lemma 3 we can remove a suitable vertex with number, say, N + 1, such that the remaining subsystem H N is still connected. Further we may assume that H N is H-integrable, since otherwise it would contain a 4-chain by the induction assumption. The coupling between H N and H N+1 is neither uniform nor zero, since then H would be integrable by Proposition 3 or disconnected. Hence we may decompose H N into a maximal subsystem H 0N which is not coupled with H N+1 and the remainder H 1N which is uniformly coupled with H N+1 . Both subsystems are non-empty and H-integrable by Theorem 1. H

HN+1

H1 N

N H0 N

36

R. Steinigeweg, H.-J. Schmidt

H N is connected and H-integrable and hence, by Theorem 2, the uniform union of two non-empty subsystems H A and H B . Both subsystems are further decomposed into H 0A , H 1A and H 0B , H 1B according to their coupling with H N+1 , similarly as H N above. Let A0 , A1 , B0 , B1 be the corresponding subsets of {1, . . . , N}. A0 ∪ B0 = ∅ and A1 ∪ B1 = ∅, see above.

HA

HN+1

1 HA

0 HA

1 HB

0 HB HB

We consider the case A0 = ∅. This means that H N+1 as well as H B is uniformly coupled to H A . The restriction of H to {N + 1} ∪ B cannot be H-integrable, since then H would be H-integrable by Proposition 3. Hence {N + 1} ∪ B contains a 4-chain by the induction assumption. Analogously we can argue in the case B0 = ∅. Hence we may assume A0 = ∅ and B0 = ∅. Assume that H N is a pantahedron. Then H would be the uniform union of H 1N and the disjoint union of H N+1 and H 0N and hence H-integrable by Proposition 3, contrary to previous assumptions.

H1 N

H N+1

H0 N

Thus H is not a pantahedron and possesses at least one uncoupled pair of spins, say (μ, ν) such that Jμν = 0. We have μ ∈ A0 , ν ∈ A1 or μ ∈ B0 ,

Integrable Spin Systems

37

ν ∈ B1 , since A and B are uniformly coupled. Because of A0 = ∅ and B0 = ∅ we can choose any λ ∈ A0 or λ ∈ B0 and obtain a 4-chain (N + 1, ν, λ, μ). HA

HN+1

1 HA

0 HA

1 HB

0 HB HB

 

5 Time Evolution of B-partitioned Systems 5.1 Explicit Form In this section we consider spin systems which are the uniform or disjoint union of subsystems A and B in such a way that these subsystems enjoy the same property, and so on, until one or both subsystems consist of single spins. Thus we obtain a nested system of partitions which can be encoded in a binary partition tree B . Examples are H-integrable spin graphs which are special B -partitioned systems by virtue of Theorem 2. The time evolution of such systems can be exactly described by means of a recursive procedure, see [19]. In this section we will, additionally, provide an explicit formula for the general solution of the equations of motion which was assumed to be “too cumbersome” in [19], using the notion of a “partition tree”. Definition 2 A partition tree B over a finite set {1, . . . , N} is a set of subsets of {1, . . . , N} satisfying 1. ∅ ∈ / B and {1, . . . , N} ∈ B , 2. for all M, M ∈ B either M ∩ M = ∅ or M ⊂ M or M ⊂ M, ˙ 2. 3. for all M ∈ B with |M| > 1 there exist M1 , M2 ∈ B such that M = M1 ∪M ˙ 2 in It follows from 2(ii) that the subsets M1 , M2 satisfying M = M1 ∪M definition 2(iii) are unique, up to their order. M1 , M2 are hence defined for all M ∈ B with |M| > 1. M1 and M2 denote the two uniquely determined “branches” starting from M. It follows that B is a binary tree with the root {1, . . . , N} and singletons {μ} as leaves. More general partitions into k disjoint

38

R. Steinigeweg, H.-J. Schmidt

subsets can be reduced to subsequent binary partitions and hence need not be considered. For all M ∈ B there is a unique path P M (B ) ≡ {M ∈ B | M ⊂ M }

(42)

joining M with the root of B . It is linearly ordered since M ⊂ M and M ⊂ M imply M ⊂ M or M ⊂ M by Definition 2 (ii). Especially, every element μ ∈ {1, . . . , N} belongs to a unique, linearly ordered construction path Pμ (B ) ≡ {M ∈ B | μ ∈ M} .

(43)

A partition tree B is called monotone, iff it contains, besides the singletons {μ}, all ordered subsets of {1, 2, . . . , N} of the form {1, 2, . . . , n}, n ≤ N. Although there are many different partition trees over a fixed finite set, all have the same size which can be easily determined by induction over N: Proposition 4 |B | = 2N − 1 . For M = {1, . . . , N} we will denote by M the “successor” of M, that is, the smallest element of P M (B ) except M itself. To simplify later definitions we set {1, . . . , N} ≡ 0 and denote by B the class of all successors, i. e. B ≡ {M | M ∈ B } .

(44)

It follows that |B | = N. Later we will use B as an index set for N action variables. For μ = ν ∈ {1, . . . , N} let Mμν ∈ B denote the smallest set of B such that μ, ν ∈ Mμν , i. e. Mμν ∈ B is the set where both construction paths of μ and ν meet the first time. Consider real functions J defined on a partition tree J : B −→ R .

(45)

Then H=



J(Mμν )sμ · sν

(46)

μ<ν

defines a Heisenberg Hamiltonian. The corresponding spin system will be called a B -partitioned system or sometimes, more precisely, a (B , J)− system. The N-pantahedron is a (B , J)-system where B is arbitrary and J is a constant function. As mentioned before, all H-integrable spin graphs are (B , J)-systems. For example, the spin square is obtained by the partition tree B = {{1, 2, 3, 4}, {1, 3}, {2, 4}, {1}, {2}, {3}, {4}}

(47)

and the function J with J({1, 2, 3, 4}) = 1 and J(M) = 0 else. Recall that S M denotes the total spin vector of the subsystem M ⊂ {1, . . . , N} with length S M and that Ft (H) denotes the flow map describing

Integrable Spin Systems

39

time evolution of spins according to a Hamiltonian H. It follows by Lemma 6 that the squares of the total spins corresponding to a partition tree commute:  2  S M , S2M = 0 for all M, M ∈ B . (48) We consider rotations in 3-dimensional spin space: Definition 3 D(ω,  t) will denote the rotation matrix with axis ω  and angle |ω|  t.  t) equals the identity matrix I. D(0, The proof of the following proposition is easy and will be omitted: Proposition 5 Let M, M ⊂ {1, . . . , N}. Then   1. Ft 12 S2M = D( S M , t) , 2. D( S M , t) S M = S M if M ⊂ M or M ∩ M = ∅ . A short calculation shows that the Hamiltonian (46) of a (B , J)-system can also be written as  1     1  J(M) − J M S2M , H= J(M) S2M − S2M1 − S2M2 = (49) 2 M∈ B 2 M∈ B if we set J({μ}) = J(0) = 0 for all μ ∈ {1, . . . , N}. It follows that

   2    1 J(M) − J M S M = Ft (H) = Ft D S M , J(M) − J M t , 2 M∈ B M∈ B (50) where the product is understood as the composition of flow maps or rotations and the order of the composition does not matter according to (48) and Lemma 1. In order to calculate Ft (H)sμ we need only consider those factors in the product (50) where μ ∈ M, i. e. M ∈ Pμ (B ). This follows by Proposition 5(ii). Moreover, if we choose a decreasing sequence of sets from left to right in the product (50), we can write D( S M , (J(M) − J(M))t) = D( S M (0), (J(M) − J(M))t). Hence we have proven the following: Theorem 4 Let H be a (B , J)-system. Then its time evolution can be written in the form ←    sμ (t) = D S M (0), J(M) − J M t sμ (0) , (51) M∈Pμ (B)

where the arrow above the product symbol denotes a product according to a decreasing sequence of sets M ∈ Pμ (B ) from left to right. We note without proof that the time evolution in the presence of a Zeeman  where B  · S,  is the dimensionless term in the Hamiltonian of the form H + B  t). magnetic field, is obtained by multiplying (51) from the left with D( B,

40

R. Steinigeweg, H.-J. Schmidt

5.2 Action-angle Variables Although Theorem 4 gives a final answer to the problem of the time evolution of a (B , J)-system, it will be yet instructive to relate this result to the general form of the time evolution in terms of action-angle variables. We will give  · S = Be · S in the the result for the general case including a Zeeman term B Hamiltonian but without detailed proofs.    We write S M =  1 S2Me M for M ∈ B . For 1M2= {1, . . . , N} = 0 we set e0 = e   and S0 = e · S. Ft 2 S M = D( S M , t) and { f, 2 S M } = { f, S M }S M imply Ft (S M ) = D(e M , t). Thus the functions S M , M ∈ B , generate rotations about the axes eM with unit angular velocity. Moreover, due to Proposition 5 they represent N commuting constants of motion and hence are good candidates for action variables. It remains to define suitable angles which change with unit angular velocity only under rotations about the axes eM . To this end consider the three unit vectors eM , eM , and eM1 . We may assume eM = ±e M and eM = ±e M1 since the action-angle variables need only be defined on an open dense subset of the phase space. Defining   eM × eM × eM   eθ ≡  (52) eM × eM × e  M and eϕ ≡ eM × eθ

(53)

we obtain an orthonormal frame (eθ , eϕ , eM ) depending on the chosen point in phase space. We define polar and azimuthal angles θ M , ϕ M by expanding eM1 into this basis: eM1 = sin θ M cos ϕ M eθ + sin θ M sin ϕ M eϕ + cos θ M eM .

(54)

Then the following result holds: Proposition 6 The functions S M , ϕ M , M ∈ B, define action-angle variables of a (B , J)-system. Especially, they satisfy {ϕ M , S M } = δ M M for all M, M ∈ B .

(55)

5.3 The Quantum Case A given (B , J)-system and a spin quantum number s ∈ { 12 , 1, 32 , . . .} uniquely specify the corresponding spin system. All functions on phase space considered above have unique representations as Hermitean operators acting upon a (2s + 1) N -dimensional Hilbert space H. To avoid complications we postulate a correspondence between functions on phase space and operators only for those functions which are sums of monomials of siμ with Poisson-commuting factors. Usually, this correspondence will be denoted by a sub-tilde,

Integrable Spin Systems

41

e. g. ∼ S M , , M ∈ B. The commutator of operators corresponds to the Poisson bracket of the corresponding functions, setting  = 1:     f , g = i f, g . (56) ∼ ∼

∼

Hence for B -partitioned systems the N operators ∼ S2M , M ∈ B , commute too. Together with S(3) ≡ e · S, where e is the unit vector into the direction of ∼

∼

the magnetic field, these operators constitute a complete system, that is, their common eigenvectors are uniquely determined by the corresponding eigenvalues S2 ∼M

= S M (S M + 1) ,

S(3) = S(3) ,

∼

(57) (58)

and span H. Hence we may write = |(S M ) M∈ B  .

(59)

Since the Hamiltonian is a function of the ∼ S2M , ∼ S(3) , = H ∼

  1   J(M) − J M S2M + BS(3) , ∼ ∼ 2 M∈ B

(60)

and the corresponding eigenvalues the vectors (59) are also eigenvectors of H ∼ can be read off from (60). The eigensystem { } can be obtained by an multiple tensor product representation of SU(2) starting from the (2s + 1)-dimensional irreducible representations corresponding to single spins and following the sequential partition of {1, . . . , N} given by B . The product representation of two irreducible representations corresponding to quantum numbers S1 , S2 is spanned by vectors of the form

        (3) (3) (3) (3)   S, S(3) = CG S1 , S2 , S(3)  S1 , S1 ⊗  S2 , S2 , (61) 1 , S2 ; S, S (3) S(3) 1 , S2

where CG(. . .) denotes the Clebsch-Gordon coefficients, see, for example, [26] 27.9.1. In order to write the product representation in compact form we introduce the notations B ≡ {M ∈ B | M = {1, . . . , N}} ,

(62)

B1 ≡ {M ∈ B | |M| > 1 } .

(63)

= |(S M ) M∈ B 

(64)

Then can be written as

=

 (S(3) M ) M∈ B

M∈ B1

   (3) (3) (3) (3) S . CG S M1 , S M2 , S(3) , S ; S , S , . . . , S  M 1 M1 M2 M N

(65)

42

R. Steinigeweg, H.-J. Schmidt

(3) Here |S(3) 1 , . . . , S N  denotes the product state which is a common eigenvector (3) of all ∼ Sμ with eigenvalues S(3) μ , μ = 1, . . . , N. This concludes the diagonalization of (B , J)-systems in the quantum case.

6 Summary and Outlook We have completely characterized H-integrable spin graphs by the graph theoretical property not to contain 4-chains. For these spin graphs as well as for the larger class of B -partitioned systems the time evolution can be explicitly calculated as a product of certain rotations about constant axes. But obviously this property is very rare. It means that four connected spins must not form a chain but rather close to build a triangle or a square. Hence this special type of integrability will either be satisfied for the majority of small spin graphs consisting of, say, four or five spins, see the Appendix. Or it may be satisfied for larger systems which are close to the pantahedron type, where each spin is uniformly coupled to each other one. As a rule, spin lattices do not fall into this class. Possible direct applications of our theory will thus be confined to small clusters or magnetic molecules. There is, however, another possible application of H-integrable spin graphs to the construction of numerical integrators for non-integrable spin systems. Numerical integrators can be viewed as computable transformations of phase space which approximate the exact time evolution. It appears that those transformations which respect the symplectic structure of phase space or, equivalently, its Poisson bracket lead to numerical integrators with favorable properties, see [27]. They are called “symplectic integrators”. The splitting of a non-integrable Heisenberg Hamiltonian into integrable parts gives rise to symplectic integrators by means of Suzuki-Trotter decompositions, which exist of various orders, see [28]. Such a splitting is always possible, just consider the decomposition into dimer Hamiltonians. But we expect that the numerical integrator is the better the larger the parts of the splitting are. In this situation the present results on H-integrable spin graphs including the explicit form of the time evolution can be utilized to develop efficient symplectic integrators for non-integrable spin systems. We have obtained first results in this direction [13].

Appendix: Connected Spin Graphs with N ≤ 5 Table 1 contains all connected spin graphs up to N = 5 spins. We have given a maximal number of independent, commuting constants of motion, using the abbreviation sij2 = si · s j. For N = 5 the constant of motion S2 is not explicitely mentioned. For H-integrable spin graphs a decomposition into uniformly coupled subsystem is indicated. If the system is not H-integrable the decomposition shows a sub-chain of length 4.

Integrable Spin Systems

43

Table 1 Spin graphs up to N = 5 spins N 1

System

Constants of motion

1

2

1

2

3

1

2

3

S (3)

H-integrable Decomposition yes

H, S (3)

yes

1

2

H, S (3) , s 213

yes

1

2

H, S (3) , s 213

yes

H, S (3) , s 213 , s 224

yes

H, S (3) , s 213 , s 224

yes

H, S (3) , s 213 , s 224

yes

H, S (3) , s 234 , s 213 + s 214

yes

1

1

3

2

4

1

3

2

4

1

3

2

4

1

3

2

4

3

2

4

1

3

2

4

1

3

2

4

1

3

2

1 4

1

2

3

4

2

1

H, S (3) , s 234 , s 213 + s 214

2 3

1

2

3

4

3

no

H, S (3) , s 213 + s 214 + s 224

4

2

yes

5

5

4

1

3

2

H, S (3) , s 213 , s 224 + s 215 + s 225 + s 235

no

H, S (3) , s 213 , s 224 + s 215 + s 225 + s 235

no

H, S (3) , s 213 , s 212 + s 223 , s 215 + s 225 + s 235

yes

H, S (3) , s 213 , s 212 + s 223 , s 215 + s 225 + s 235

yes

5

4

1

3

2

5

4

1

3

2

5

4

1

3

2

5

4

1

3

2

4

1

3

2

5

5 4

1

3

2

1

2

3

4

5

H, S (3) , s 213 + s 214 + s 215 + s 224 + s 225 + s 235

3 1

2

4

5

4

1

3

2

no 1

H, S (3) , s 213 + s 214 + s 215 + s 225 + s 235

no

H, S (3) , s 213 + s 214 + s 215 + s 224 + s 225

no

2

3

1

4

2

S (3) ,

s 213

+

s 214

+

s 215

+

s 224

+

s 225

5

1 2

5 3

H, S (3) , s 213 + s 214 + s 224 + s 225 + s 235

no

H, S (3) , s 212 , s 234 , s 213 + s 214 + s 223 + s 224

yes

4

1 2

5 4

2

3

3

no

4

5

5

4

2

H,

3

1

3

1

2

4

4

2 5

1

5

4

1

2 5

3

3

1

4

3

1

4

5

3

3

1

2

5 4

3

5

44

R. Steinigeweg, H.-J. Schmidt

Table 1 (continued) 1

1

2

H, S (3) , s 212 , s 234 , s 213 + s 214 + s 223 + s 224

5 4

3

1

2

5 4

3

1

5 4

2

yes

3

H, S (3) , s 212 , s 234 , s 213 + s 214 + s 223 + s 224

yes

H, S (3) , s 213 + s 224 + s 225 + s 235

no

H, S (3) , s 213 + s 225 + s 235 , s 214 + s 224 + s 234 + s 245

no

H, S (3) , s 214 , s 223 , s 225 + s 235

yes

H, S (3) , s 213 , s 224 , s 225 + s 245

yes

2

5 4

3

5

5

4

1

3

2

4

1

3

2

5

5

4

1

3

2

4

1

3

2

5

5

4

1

3

2

4

1

3

2

5

5

1

4 2

3

2

3

5

5

1

4

H, S (3) , s 213 , s 224 , s 225 + s 245

yes

2

3

2

3

5

1

H, S (3) , s 213 , s 224 , s 225 + s 245

yes

2

2

3

5

1

H, S (3) , s 213 , s 225 , s 224 + s 245

yes

2

2

3

5

1

H, S (3) , s 213 , s 225 , s 224 + s 245

yes

2

2

3

5

1

H, S (3) , s 213 , s 224 , s 215 + s 225 + s 235 + s 245

yes

2

2

3

5

1

H, S (3) , s 213 , s 224 , s 215 + s 225 + s 235 + s 245

yes

2

1

4 2

3

5

5

1

4 3

1

4

5

4 3

1

4

5

4 3

1

4

5

4 3

1

4

5

4 3

1

4

5

4 3

1

4

2

H, S (3) , s 213 , s 224 , s 215 + s 225 + s 235 + s 245

yes

1

4 3

2

References 1. Magyari, E., Thomas, H., Weber, R., Kaufman, C., Müller, G.: Integrable and nonintegrable classical spin clusters: integrability criteria and analytic structure of invariants. Z. Phys. B 65, 363 (1987) 2. Srivastava, N., Kaufman, C., Müller, G., Weber, R., Thomas, H.: Integrable and nonintegrable classical spin clusters: trajectories and geometric structure of invariants. Z. Phys. B 70, 251 (1988) 3. Santos, L.F.: Integrability of a disordered Heisenberg spin-1/2 chain. J. Phys. A, Math. Gen. 37, 4723 (2004)

Integrable Spin Systems

45

4. Finkel, F., Gonzáles-López, A.: Title: global properties of the spectrum of the Haldane-Shastry spin chain. Phys. Rev. B 72, 174411 (2005) 5. Casati, G., Ford, J., Vivaldi, F., Visscher, W.M.: One-dimensional classical many-body system having a normal thermal-conductivity. Phys. Rev. Lett. 52, 1861 (1984) 6. Li, B., Casati, G., Wang, J., Prosen, T.: Fourier law in the alternate-mass hard-core potential chain. Phys. Rev. Lett. 92, 254301 (2004) 7. Steinigeweg, R., Gemmer, J., Michel, M.: Normal-transport behavior in finite one-dimensional chaotic quantum systems. Europhys. Lett. 75, 406 (2006) 8. Gatteschi, D.: Molecular magnetism - a basis for new materials. Adv. Mater. 6, 635 (1994) 9. Winpenny, R.E.P.: Families of high nuclearity cages. Comment. Inorg. Chem. 20, 233 (1999) 10. Müller, A., Peters, F., Pope, M.T., Gatteschi, D.: Polyoxometalates: very large clustersnanoscale magnets. Chem. Rev. 98, 239 (1998) 11. Gatteschi, D., Sessoli, R., Cornia, A.: Single-molecule magnets based on iron(III) oxo clusters. Chem. Commun. 9, 725 (2000) 12. Bencini, A., Gatteschi, D.: Electron Parametric Resonance of Exchange Coupled Systems. Springer, Berlin (1990) 13. Steinigeweg, R., Schmidt, H.-J.: Symplectic integrators for classical spin systems. Comput. Phys. Comm. 147(11), 853 (2006) 14. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978) 15. Skylanin, E.K.: In: Mo-Lin, G. (ed.) Quantum Groups and Quantum Integrable Systems, pp. 63–97. World Scientific, Singapore (1992) 16. Schröder, Ch.: Numerische Simulation zur Thermodynamik magnetischer Strukturen mittels deterministischer und stochastischer Wärmebadankopplung, Dissertation, Universität Osnabrück (1999) 17. Richter, J., Voigt, A.: The spin-1/2 Heisenberg star with frustration - numerical versus exact results. J. Phys. A, Math. Gen. 27, 1139 (1994) 18. Richter, J., Voigt, A., Krüger, S.: The spin-1/2 Heisenberg star with frustration: the influence of the embedding medium. J. Phys. A, Math. Gen. 29, 825 (1996) 19. Ameduri, M., Gerganov, B., Klemm, R.A.: Classification of integrable clusters of classical Heisenberg spins. Preprint cond-mat/0502323 (2008) 20. Ballestros, A., Ragnisco, O.: A systematic construction of completely integrable Hamiltonians from coalgebras. J. Phys. A, Math. Gen. 31, 3791 (1998) 21. Gaudin, M.: La Fonction d’Onde de Bethe. Mansson, Paris (1983) 22. Calogero, F.: An exactly solvable Hamiltonian system - quantum version. Phys. Lett. A 36, 306 (1995) 23. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1999) 24. Abraham, R., Marsden, J.E., Ratiu, T.S.: Manifolds, Tensor Analysis, and Applications. Addison-Wesley, London (1983) 25. Swamy, M.N.S., Thulasiraman, K.: Graphs, Networks, and Algorithms. Wiley, New York (1981) 26. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1965) 27. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, New York (2002) 28. Tsai, S., Krech, M., Landau, D.P.: Symplectic integration methods in molecular and spin dynamics simulations. Braz. J. Phys. 34(2A), 384 (2004)

Math Phys Anal Geom (2009) 12:47–74 DOI 10.1007/s11040-008-9051-x

Homogeneous Quaternionic Kähler Structures on Eight-Dimensional Non-Compact Quaternion-Kähler Symmetric Spaces M. Castrillón López · P. M. Gadea · J. A. Oubiña

Received: 19 September 2007 / Accepted: 14 November 2008 / Published online: 13 December 2008 © Springer Science + Business Media B.V. 2008

Abstract For each non-compact quaternion-Kähler symmetric space of dimension eight, all of its descriptions as a homogeneous Riemannian space, and the associated homogeneous quaternionic Kähler structures obtained through the Witte’s refined Langlands decomposition, are studied. Keywords Homogeneous quaternionic Kähler structures · Homogeneous Riemannian structures · Non-compact quaternion-Kähler symmetric spaces · Non-linear σ -models in N = 2 supergravity · Parabolic subgroups · Refined Langlands decomposition Mathematics Subject Classifications (2000) 53C26 · 53C30 · 53C35 1 Introduction and Preliminaries 1.1 Introduction Quaternion-Kähler symmetric spaces were classified by Wolf [21]. The noncompact ones are moreover a particular case of Alekseevsky spaces [1, 8]. On

M. Castrillón López (B) Departamento de Geometría y Topología, Facultad de Matemáticas, Av. Complutense s/n, 28040, Madrid, Spain e-mail: [email protected] P. M. Gadea Institute of Fundamental Physics, CSIC, Serrano 113-bis, 28006, Madrid, Spain e-mail: [email protected] J. A. Oubiña Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain e-mail: [email protected]

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the other hand, a theoretical-representation classification of homogeneous quaternionic Kähler structures was given by Fino [9, L. 5.1]. Then, a classification by real tensors was obtained in [6, Th. 3.15], where furthermore a characterization of the quaternionic hyperbolic space in terms of such structures was given. The aim of this paper is to find all the homogeneous descriptions of each 8-dimensional non-compact quaternion-Kähler symmetric space, the associated homogeneous Riemannian structures obtained through Witte’s refined Langlands decomposition [20], and finally to determine the types of these structures as homogeneous quaternionic Kähler structures. Non-compact quaternion-Kähler spaces are found in the formulation of the coupling of matter fields in N = 2 supergravity as target spaces of some nonlinear sigma models (Bagger-Witten [4], Cecotti [7], de Wit-Van Proeyen [19]; see also Fré et al. [10] and references therein). Since Fino’s classification is natural, it seems reasonable the existence of links between the classification of homogeneous quaternionic Kähler structures and a possible classification of some physical structures and models. This would, in a sense, be similar to a recent result by Meessen [16]: A connected homogeneous Lorentzian space admits a non-vanishing degenerate homogeneous Lorentzian structure in Tricerri-Vanhecke’s class T1 ⊕ T3 (see [11, 17, 18]) if and only if it is a singular homogeneous plane wave. Moreover, we give in the present paper some candidates to such target manifolds. As for the contents, we first establish (Theorem 2.3) a result related to Witte’s Theorem on closed co-compact subgroups of a semisimple Lie group with finite center [20], which gives necessary and sufficient conditions for a connected closed co-compact subgroup of such a group to act transitively (see also Gordon and Wilson [12]). We then furnish explicitly (Proposition 2.4), for the present case of symmetric spaces, the coefficients appearing in the symmetries of homogeneous structures characteristic of the quaternionic Kähler case (that is, Eq. 1.3). Further, for each one of spaces under study, i.e. the complex hyperbolic Grassmannian SU(2, 2)/S(U(2) × U(2)), the quaternionic hyperbolic space HH(2) = Sp(2, 1)/(Sp(2) × Sp(1)), and the exceptional space G2(2) /SO(4), we first give the corresponding quaternionic structure. This is associated to a natural structure of quaternion-Hermitian vector space on the Lie algebra a + n of the solvable factor AN of an Iwasawa decomposition of its respective full connected group of isometries. According to Alekseevsky [1] (see also Heber [13]), its metric is unique up to isometry and scaling. Then, we obtain all the descriptions of each space and the corresponding homogeneous quaternionic Kähler structures. These structures are associated to the reductive decompositions defined via the Witte’s refined Langlands decomposition of the identity component of each parabolic subgroup of the full connected isometry group. It turns out that each parabolic subgroup provides a homogeneous description which, although is the simplest possible one, suffices to obtain all such homogeneous structures. We exhibit the corresponding structures in Propositions 3.1, 3.2, 3.3, and their types in Theorem 3.4. We have studied the twelve-dimensional case in [5].

Homogeneous Quaternionic Kähler Structures

49

1.2 Homogeneous Quaternionic Kähler Structures Let (M, g) be a connected, simply-connected, and complete Riemannian manifold. Ambrose and Singer [3, p. 656] gave a characterisation for (M, g) to be homogeneous in terms of a (1, 2) tensor field S, usually called (Tricerri and Vanhecke [18]) a homogeneous Riemannian structure: Let ∇ be the LeviCivita connection of g and R its curvature tensor. Then the manifold is homogeneous Riemannian if and only if there exists a (1, 2) tensor field S such that the Ambrose-Singer equations  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

(1.1)

 = ∇ − S, are satisfied. where ∇ Let (M, g, υ) be a quaternion-Kähler manifold, υ denoting the distinguished rank-three subbundle of the bundle of (1, 1) tensor fields on M. Then (M, g, υ) is said to be a homogeneous quaternion-Kähler space if it admits a transitive group of isometries (Alekseevsky and Cortés [2, Th. 1.1]). The canonical 4-form  is known to be globally defined and we have Corollary 1.1 (To Kiri˘cenko [14, Th. 1]) A connected, simply-connected, and complete quaternion-Kähler manifold (M, g, υ) is homogeneous if and only if there exists a tensor field S of type (1, 2) on M satisfying  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 = 0, ∇

(1.2)

 = ∇ − S. where ∇ Such a tensor S is called a homogeneous quaternionic Kähler structure on M.  = 0 can be replaced by the set of equations The condition ∇ S X J1 = θ 3 (X)J2 − θ 2 (X)J3 , S X J2 = θ 1 (X)J3 − θ 3 (X)J1 , S X J3 = θ 2 (X)J1 − θ 1 (X)J2 ,

(1.3)

for certain differential 1-forms θ a , a = 1, 2, 3, where {J1 , J2 , J3 } is a local basis of υ satisfying the conditions Ja2 = −I, Ja Jb = −Jb Ja = Jc , for each cyclic permutation (a, b , c) of (1, 2, 3). Denoting S XY Z = g(S X Y, Z ), one has S X J1 Y J1 Z − S XY Z = θ 3 (X)g(J2 Y, J1 Z ) − θ 2 (X)g(J3 Y, J1 Z ),

(1.4)

and the two other similar equations. Note that the conditions (1.4) (equivalently, Eq. 1.3), besides the general condition S XY Z = −S X Z Y , are the symmetries satisfied by a homogeneous quaternionic Kähler structure S.

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A quaternion-Hermitian real vector space (V,  , , J1 , J2 , J3 ) is the model for the tangent space at any point of a quaternion-Kähler manifold. Consider the space of tensors T (V) = {S ∈ ⊗3 V ∗ : S XY Z = −S X Z Y } and its vector subspace V of tensors satisfying (1.4) with  , , θ a ∈ V ∗ . Then V = Vˇ + Vˆ , where   3 a Vˇ =  ∈ ⊗3 V ∗ :  XY Z = a=1 θ (X) Ja Y, Z  , θ a ∈ V ∗ ,   Vˆ = T ∈ ⊗3 V ∗ : T XY Z = −T X Z Y , T X Ja Y Ja Z = T XY Z , a = 1, 2, 3 . The decomposition above with respect to the scalar product 4n of V is orthogonal ( , ) given by (S, S ) = r,s,t=1 Ser es et S er es et , where {er }r=1,...,4n is an orthonormal basis of V. Then Vˇ and Vˆ decompose into two and three subspaces, respectively, giving an orthonormal sum of 5 subspaces which are invariant and irreducible under the action of Sp(n)Sp(1), due to the following theorem. Let E denote the standard representation of Sp(n) on C2n ; S3 E the 3-symmetric product of E; K the irreducible Sp(n)-module of highest weight (2, 1, 0, . . . , 0); H the standard representation of Sp(1) on C2 ; and S3 H the 4-dimensional symmetric product of H. Using brackets for real representations and with the usual notations we have Theorem 1.2 (Fino [9, L. 5.4]) The space [EH] ⊗ (sp(1) ⊕ sp(n)) of homogeneous quaternionic Kähler structures decomposes into invariant and irreducible subspaces under the action of Sp(n)Sp(1) as [EH] + [ES3 H] + [EH] + [S3 EH] + [K H]. Let QKi be the ith summand in Fino’s classification. Then we have Theorem 1.3 ([6, Th. 3.15]) If n  2, then V decomposes into the direct sum of the following subspaces invariant and irreducible under the action of Sp(n)Sp(1):   ∈ Vˇ :  XY Z =

QK1 =

3

θ(Ja X) Ja Y, Z  , θ ∈ V

a=1

 QK2 =

 ∈ Vˇ :  XY Z =

3

θ a (X) Ja Y, Z  ,

a=1 3 a=1

θ ◦ Ja = 0, θ ∈ V a

a

∗

,

∗

,

Homogeneous Quaternionic Kähler Structures

51

 T ∈ Vˆ : T XY Z = X, Yϑ(Z ) − X, Z ϑ(Y)

QK3 =

+  QK4 =

3 

X, Ja Yϑ(Ja Z ) − X, Ja Z ϑ(Ja Y) , ϑ ∈ V ∗

,

a=1

 T ∈ Vˆ : T XY Z = 1/2 TY Z X − T Z Y X 3 + (T Ja Y Ja Z X − T Ja Z Ja Y X ) , c12 (T) = 0

 QK5 =

,

a=1

 T ∈ Vˆ : T XY Z = −1/4 TY Z X − T Z Y X 3 + (T Ja Y Ja Z X − T Ja Z Ja Y X )

.

a=1

We will denote the sum of classes QKi + QK j by QKij, and so on. In particular, we have Vˇ = QK12 , Vˆ = QK345 . Remark 1.4 Let (M, g, υ) be a homogeneous quaternion-Kähler space. For any given p ∈ M, the decomposition of V p given by Theorem 1.3 depends only on υ p and not on the chosen bases {(J1 ) p , (J2 ) p , (J3 ) p }, so the irreducible summands (QKi ) p give well-defined bundles QKi over M. Moreover, if S is a homogeneous quaternionic Kähler structure and S p belongs to a given invariant subspace of V p , then Sq belongs to the invariant subspace of Vq of the same type at any other q ∈ M, and S is a section of the corresponding vector bundle.

2 Non-compact Homogeneous Spaces 2.1 Non-discrete Co-compact Subgroups Acting Transitively Gordon and Wilson gave in [12, Th. 6.9] a theorem of characterization of the isometry groups acting transitively on non-compact Riemannian symmetric spaces. We will prove Theorem 2.3, related to Witte’s Theorem 2.2, which suffices for our purposes. Let (M, g) be a connected non-compact Riemannian manifold and G a Lie group of isometries acting transitively and effectively on M. If K is the isotropy group at any fixed point o ∈ M, then M = G/K. We look for the ˆ of G acting transitively by isometries on M. connected closed subgroups G ˆ ∩ K, one must have M ≡ G/K ˆ Denoting KGˆ = G ˆ , and thus other possible G

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description of (M, g) as a homogeneous Riemannian space. Note that since G ˆ then G/G ˆ ≡ K/K ˆ . Then acts effectively then K is compact, and if G = GK G we have Lemma 2.1 ˆ of G acts transitively on M if and only if G = GK. ˆ (1) A subgroup G ˆ (2) If G is a closed subgroup of G which acts transitively on M, then it is ˆ is compact. co-compact, that is, G/G The structure of the non-discrete co-compact subgroups of a connected semisimple Lie group G with finite center was given in Witte [20, Th. 1.2]: Let g be the Lie algebra of G, a a maximal R-diagonalizable subalgebra of g,  the set of roots of (g, a), and g = g0 + f ∈ g f the restricted-root space decomposition, with g0 = a + Z k (a), where Z k (a) stands for the centralizer of a in k. Write  + for the set of positive roots with respect to a certain notion of positivity for a∗ , and let be the set of simple restricted roots. For each subset

of , let [ ] be the set of restricted roots that are linear combinations of ele0 ments of . Then, the (connected) standard parabolic subgroup  P is defined as the connected subgroup of G having Lie algebra p = g0 + f ∈ + ∪[ ] g f =  l + n , where l = g0 + f ∈[ ] g f = l + e + a , with l semisimple with noncompact summands, e compact reductive, a the non-compact part of the  center of l , and with n = f ∈ + \[ ] g f nilpotent. On the Lie group level we 0 have [20, Th. 1.2] the refined Langlands decomposition P

= L E

A N , and one has the Theorem 2.2 (Witte [20, Th. 1.2]) Let L be a connected normal subgroup of L

and E a connected closed subgroup of E

A . Then there is a closed co-compact ˆ ˆ 0 = LEN . subgroup G of G contained in P with identity component G Further, every closed co-compact subgroup of G arises in this way. If I0 (M) is the full connected isometry group of M, to find the homogeneous descriptions of the non-compact Riemannian symmetric space (M, g), we ˆ of I0 (M) given by Witte’s will look among the co-compact subgroups G Theorem 2.2 for those connected groups acting transitively. Note that if a Lie group acts transitively on M, so does its identity component. From Lemma 2.1 and Theorem 2.2, and writing N = N∅ when = ∅ ⊂ , we have Theorem 2.3 Let G be a connected semisimple Lie group with finite center and G = K AN the Iwasawa decomposition. A connected closed co-compact ˆ = LEN of G acts transitively on M = G/K if and only if: (a) subgroup G   The projections of the Lie algebra l ⊂ l = g0 + f ∈[ ] g f of L to f ∈ + ∩[ ] g f ⊥ and to a⊥

are surjective, a being the orthogonal complement to a in a ⊂ g0 = ⊥ a + a +Z k (a). (b) The projection of the Lie algebra e ⊂ e + a of E to a is surjective.

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ˆ = G. Since G ˆ and K are connected Proof Transitivity is equivalent to GK ˆ subgroups of G, this is equivalent to saying that the Lie algebra gˆ + k of GK coincides with g. On the other hand, the Iwasawa decomposition of G gives us the vector space direct sum decomposition g = k + a + n. Then, if πk , πa , and πn denote the projections to the summands k, a and n, a necessary and sufficient condition for gˆ + k = g to be fulfilled, = a and πn (ˆg) = n.  is to have πa (ˆg) Hence we have that: (1) Since n = f ∈ + g f and n = f ∈ + \[ ] g f , to get πn (ˆg) = n, the first condition in (a) must be true, and (2) To have πa (ˆg) = a, as A and on the other hand a ⊂ a, the on the one hand E is a subgroup of E

other condition in (a) and the one in (b) must be satisfied.   2.2 Homogeneous Riemannian Structures Let (M, g, υ) a connected simply-connected homogeneous quaternion-Kähler space. We have M = G/K, where G = I0 (M) and K is the isotropy subgroup ˆ be a connected closed Lie subgroup of G which of G at a point o ∈ M. Let G acts transitively on M. The isotropy group of this action at o = K ∈ M is ˆ ∩ K. Then M = G/K has also the description M ≡ G/H, ˆ H = KGˆ = G and ˆ o ≡ H ∈ G/H. ˆ that is, Consider a reductive decomposition of the Lie algebra gˆ of G, a vector space direct sum gˆ = h + m, where h is the Lie algebra of H and Ad(H) m ⊂ m. Since H is connected, the condition Ad(H) m ⊂ m is equivalent to [h, m] ⊂ m. For each X in the Lie algebra g of G, denote by X ∗ the Killing vector field on M generated by the one-parameter subgroup { exp tX } of G acting on M. The vector subspace m of gˆ ⊂ g is identified with the tangent space To (M) by the isomorphism X ∈ m → Xo∗ ∈ To (M). The reductive decompo sition gˆ = h + m defines the homogeneous Riemannian structure S = ∇ − ∇, ˆ  where ∇ is the canonical connection of M = G/H with respect to gˆ = h + m. ˆ it is determined by  is invariant under the action of G, The connection ∇ X ∗ Y ∗ )o = [X ∗ , Y ∗ ]o = − [X, Y]∗o , for X, Y ∈ m, and the Ambrose-Singer (∇ equations (1.1) are satisfied. The tensor field S is also uniquely determined ˆ ˆ by its value at o because M ≡ G/H and S is G-invariant. We have (S X ∗ Y ∗ )o = (∇ X ∗ Y ∗ )o + [X, Y]∗o = ∇Yo∗ X ∗ ,

X, Y ∈ m .

(2.1)

ˆ  = Moreover, as  is G-invariant, from [15, Ch. 10, Prop. 2.7] it follows that ∇ 0, so Eq. 1.2 are satisfied and S is also a homogeneous quaternionic Kähler structure. Now, suppose that (M = G/K, g, υ) is a connected non-compact quaternionKähler symmetric space, and let gˆ = h + m be a reductive decomposition ˆ corresponding to M ≡ G/H as above. We consider a Cartan decomposition g = k + p of the Lie algebra g of G, and the Iwasawa decomposition g = k + a + n, where k is the Lie algebra of K, a ⊂ p is a maximal R-diagonalizable subalgebra of g, and n is a nilpotent subalgebra. Let A and N be the connected abelian and nilpotent Lie subgroups of G whose Lie algebras are a and n,

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respectively. The solvable Lie group AN acts simply transitively on M, so M is isometric to AN equipped with the left-invariant Riemannian metric defined by the scalar product induced on a + n ∼ = g/k ∼ = p by a positive multiple of B| p × p , where B is the Killing form of g. If X ∈ g = k + p, we write X = Xk + Xp , (Xk ∈ k, Xp ∈ p). We have the isomorphisms of vector spaces p∼ = g/k ∼ = gˆ / h ∼ =m∼ = To (M) ∼ = a + n,

(2.2)

with ∼ =

∼ =

ξ : p → m,

μ : m → To (M),

∼ =

ζ : To (M) → a + n,

such that ξ −1 (Z ) = Z p ,

μ(Z ) = Z o∗ ,

ζ −1 (X) = Xo∗ ,

Z ∈ m, X ∈ a + n .

For each X ∈ g, we have (Xk )∗o = 0 and (∇(Xp )∗ )o = 0, and since the LeviCivita connection ∇ has no torsion, for each X, Y ∈ g, we have  ∗

(∇ X ∗ Y ∗ )o = (∇(Xp )∗ (Yk )∗ )o = (Xp )∗ , (Yk )∗ o = − Xp , Yk o . (2.3) By (2.1) and (2.3), the homogeneous Riemannian structure S associated to a reductive decomposition gˆ = h + m is given by

∗ S Xo∗ Yo∗ = Xk , Yp o , X, Y ∈ m . Then, for each X, Y ∈ a + n, we have

∗ S Xo∗ Yo∗ = Sξ(Xp )∗o ξ(Yp )∗o = (ξ(Xp ))k , Yp o .

(2.4)

This last formula will be helpful to give directly the associated homogeneous Riemannian structure S in terms of a basis of a + n. Moreover, we will get formula (2.6), which furnishes explicitly the coefficients θ a , a = 1, 2, 3, in Eq. 1.3, for the quaternion-Hermitian structure on M which we will now describe. Before giving it, we note that formulas (2.4) and (2.6) will be useful in the next sections to find the homogeneous Riemannian structures on the spaces under study and their types as homogeneous quaternionic Kähler structures, respectively. The quaternionic structure on M is defined by a 3-dimensional ideal u ∼ = sp(1) of k, and we can suppose that u =  E1 , E2 , E3 , with [E1 , E2 ] = 2E3 ,

[E2 , E3 ] = 2E1 ,

[E3 , E1 ] = 2E2 .

The next commutative diagram defines the complex structures Ja ∈ End(a + n) (1  a  3), making (a + n,  , , J1 , J2 , J3 ) a quaternion-Hermitian vector space isomorphic to (To (M), go , (J1 )o , (J2 )o , (J3 )o ). ξ

p −−−−→ ⏐ ⏐ ad Ea  ξ

μ

ζ

m −−−−→ To (M) −−−−→ a + n ⏐ ⏐ ⏐ ⏐ ⏐ ⏐J (Ja )o   a μ

ζ

p −−−−→ m −−−−→ To (M) −−−−→ a + n

(2.5)

Homogeneous Quaternionic Kähler Structures

55

Now, the Lie subgroup of K generated by u is a normal subgroup isomorphic to Sp(1), its isotropy representation is isomorphic to the representation of the normal subgroup Sp(1) of Sp(n)Sp(1) on R4n (4n = dim M), and there exists a normal subgroup K1 of K such that K = K1 Sp(1) and K1 ∩ Sp(1) is contained in the center {±1} of Sp(1). The Lie algebra k1 of K1 is an ideal of k such that k = u ⊕ k1 , and we can get a basis B = { E1 , E2 , E3 , . . . } of k with the basic elements of u and some elements of k1 . Let { α1 , α2 , α3 , . . . } be the dual basis of B . Then we have Proposition 2.4 The homogeneous Riemannian structure S on the connected non-compact quaternion-Kähler symmetric space M = G/K associated to the reductive decomposition gˆ = h + m satisfies Eq. 1.3 with 1-forms θ i , ˆ i = 1, 2, 3, given at o ≡ H ∈ G/H ≡ M by θ i (Xo∗ ) = 2αi ((ξ(Xp ))k ),

(2.6)

for each X ∈ a + n. Proof Let X, Y ∈ a + n. From the commutative diagram (2.5) we have (Ja )o Yo∗ = (ad Ea Yp )∗o . Then, from (2.4), for each a = 1, 2, 3, we get ∗

S Xo∗ ((Ja )o Yo∗ ) = (ξ(Xp ))k , ad Ea Yp o , and

 (Ja )o (S Xo∗ Yo∗ ) = (ad Ea (ξ(Xp ))k , Yp )∗o . Hence, applying Jacobi’s identity, we deduce

∗ S Xo∗ ((Ja )o Yo∗ ) − (Ja )o (S Xo∗ Yo∗ ) = − [Ea , (ξ(Xp ))k ], Yp o .

(2.7)

In terms of the basis B , we have (ξ(Xp ))k = α1 ((ξ(Xp ))k )E1 + α2 ((ξ(Xp ))k )E2 + α3 ((ξ(Xp ))k )E3 + . . . , then

 E1 , (ξ(Xp ))k = 2α2 ((ξ(Xp ))k )E3 − 2α3 ((ξ(Xp ))k )E2 .

From (2.7), for instance for a = 1, we have S Xo∗ ((J1 )o Yo∗ ) − (J1 )o (S Xo∗ Yo∗ ) ∗

= − 2α2 ((ξ(Xp ))k )E3 − 2α3 ((ξ(Xp ))k )E2 , Yp o

∗

∗ = 2α3 (ξ(Xp ))k ) E2 , Yp o − 2α2 (ξ(Xp ))k ) E3 , Yp o = 2α3 ((ξ(Xp ))k )(J2 )o Yo∗ − 2α2 ((ξ(Xp ))k )(J3 )o Yo∗ , and Eq. 1.3 are satisfied, with θ i (Xo∗ ), i = 1, 2, 3, as in the statement.

 

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3 Homogeneous Descriptions and Structures Since we want to obtain the types of homogeneous quaternionic Kähler structures on 8-dimensional non-compact quaternion-Kähler symmetric spaces, we need to get all the homogeneous descriptions of them. As they are Alekseevsky spaces, we denote by A SU(2,2) , A Sp(2,1) and AG2(2) , the spaces SU(2, 2)/S(U(2) × U(2)), HH(2) = Sp(2, 1)/(Sp(2) × Sp(1)), and G2(2) /SO(4), respectively. (For HH(2) cf. [6, Ths. 1.1, 5.4]). Since the center of each of the corresponding full isometry groups is finite, Theorems 2.2 and 2.3 apply. To find the corresponding homogeneous Riemannian structures, an important role will be played by the Lie algebra a + n of the solvable factor AN in an Iwasawa decomposition of the full connected isometry group of each space. To obtain each one of such structures S, we will use (2.4) and the identifications given by (2.2). In particular, we will also denote by X ∈ a + n the vector Xo∗ = (Xp )∗o ∈ To (M), and we will give the values S X Y = S Xo∗ Yo∗ , for all X, Y in a suitable basis of a + n. We know that S is a homogeneous quaternionic Kähler structure and that Proposition 2.4 allows us to calculate directly the forms θ a in (1.3).

3.1 The Complex Hyperbolic Space A SU(2,2) 3.1.1 The Quaternion-Hermitian Structure of a + n ⊂ su(2, 2) The Lie algebra of SU(2, 2) is  su(2, 2) =

 A B ∈ sl(4, C) : A, C ∈ u(2) . B¯ T C

¯ T defines the Cartan decomThe involution τ of su(2, 2) given by τ (X) = − X position su(2, 2) = k + p, where k = s(u(2) ⊕ u(2)). We consider the subspace a of p defined by ⎧⎛ 0 ⎪ ⎪ ⎨⎜ 0 a= ⎜ ⎝0 ⎪ ⎪ ⎩ s2

0 0 s1 0

0 s1 0 0

⎫ ⎞ s2 ⎪ ⎪ ⎬ 0⎟ ⎟ : s1 , s2 ∈ R , 0⎠ ⎪ ⎪ ⎭ 0

which is a maximal R-diagonalizable subalgebra of su(2, 2). Let A1 and A2 be the generators of a defined by (s1 , s2 ) = (1, 0) and (0, 1), respectively. Let f1 and f2 denote the linear functionals on a whose values on the above matrix of a are s1 and s2 , respectively. Then, the sets of positive roots and simple roots

Homogeneous Quaternionic Kähler Structures

57

(with respect to a suitable order in a∗ ) are given by  + = { f1 ± f2 , 2 f1 , 2 f2 } and = { f1 − f2 , 2 f2 }, respectively. The positive root vector spaces are ⎧⎛ ⎧⎛ ⎞⎫ ⎞⎫ 0 z −z 0 ⎪ 0 z −z 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎨⎜ ⎟⎬ ⎟⎬ − z ¯ 0 0 z ¯ − z ¯ 0 0 − z ¯ ⎜ ⎟ ⎜ ⎟ g f1 + f 2 = ⎝ , g f1 − f 2 = ⎝ , −z¯ 0 0 z¯ ⎠⎪ −z¯ 0 0 − z¯ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 z −z 0 0 −z z 0 (3.1)

g 2 f1

⎧⎛ 0 ⎪ ⎪ ⎨⎜ 0 = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

0 ix ix 0

⎞⎫ 0 0 ⎪ ⎪ ⎬ − ix 0 ⎟ ⎟ , − ix 0 ⎠⎪ ⎪ ⎭ 0 0

g 2 f2

⎧⎛ ix ⎪ ⎪ ⎨⎜ 0 = ⎜ ⎝ 0 ⎪ ⎪ ⎩ ix

0 0 0 0

0 0 0 0

⎞⎫ − ix ⎪ ⎪ ⎬ 0 ⎟ ⎟ , 0 ⎠⎪ ⎪ ⎭ − ix

(3.2)

where x ∈ R, z ∈ C. The root vector spaces for the respective opposite roots are the corresponding sets of opposite conjugate transpose matrices. For each f = f1 ± f2 , let X f and X f be the generators of g f in (3.1) obtained by putting z = 1 and z = i, respectively, for each non-null entry. For f = 2 f j (1  j  2), let U f be the generator of g f in (3.2) obtained by putting x = 1 in each non-zero entry. Let also X f , X f , U f , be the corresponding elements of g f for the respective opposite roots f ∈  \  + . Now, we put X1 = X f1 + f2 , Y1 = X− f1 − f2 , X1 = X f1 + f2 , Y1 = X− f1 − f2 , X2 = X f1 − f2 , Y2 = X− f1 + f2 , X2 = X f1 − f2 , Y2 = X− f1 + f2 , U 1 = U 2 f1 , V1 = U −2 f1 , U 2 = U 2 f2 , V2 = U −2 f2 . Moreover, the centralizer of a in k is generated by C = diag(i, −i, −i, i). Then, Z k (a) + a =  C,  A1 , A2  is a Cartan subalgebra of su(2, 2), and su(2, 2) = (Z k (a) + a) + f ∈ g f is the restricted-root space decomposition.  We also have the Iwasawa decomposition su(2, 2) = k + a + n, where n = f ∈ + g f =  X1 , X1 , X2 , X2 , U 1 , U 2 . The elements E1 , E2 , E3 of k = s(u(2) ⊕ u(2)) ⊂ su(2, 2) given by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 00 00 000 0 0000 ⎜0 0 0 0⎟ ⎜0 0 0 0 ⎟ ⎜0 0 0 0⎟ ⎟ ⎟ ⎟ E2 = ⎜ E3 = ⎜ E1 = ⎜ ⎝ 0 0 −i 0 ⎠ , ⎝ 0 0 0 −1 ⎠ , ⎝0 0 0 i ⎠, 00 0 i 001 0 00 i 0 satisfy [E1 , E2 ] = 2E3 , [E2 , E3 ] = 2E1 , [E3 , E1 ] = 2E2 . The compact subalgebra u ∼ = sp(1) generated by {E1 , E2 , E3 } is an ideal of k, and the isotropy representation u → gl(p) defines a quaternionic structure on A SU(2,2) . From the action of each Ea on p and through the isomorphisms p ∼ = su(2, 2)/ k ∼ = a + n, we obtain the complex structures Ja (a = 1, 2, 3), acting on a + n. The action on the elements A j, U j, X j, X j ( j = 1, 2) of the basis of a + n is the following. A1 J1

−U 1

A2 U2

U1 A1

U2

X1

−A2

X1

X1

X2

X2

−X1

X2

−X2

J2 1/2(X1−X2 ) 1/2(X1+X2 ) 1/2(X1 −X2 ) −1/2(X1 +X2 ) −A1−A2 −U 1+U 2 A1−A2 U 1+U 2 J3 1/2(X1 −X2 ) 1/2(X1 +X2 ) −1/2(X1−X2 ) 1/2(X1+X2 )

U 1−U 2 −A1−A2 −U 1−U 2 A1−A2

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Any positive multiple of the restriction of the Killing form B to p × p defines a Hermitian metric adapted to (J1 , J2 , J3 ). We consider the scalar product  ,  induced in a + n by the isomorphism p ∼ = a + n and (1/16)B| p × p . This product makes the basis orthogonal, with A j, A j = U j, U j = 1, X j, X j = X j, X j = 2, and (a + n,  , , J1 , J2 , J3 ) is a quaternion-Hermitian vector space. 3.1.2 Homogeneous Descriptions and Structures of A SU(2,2) We now determine the homogeneous descriptions and structures of A SU(2,2) . These descriptions will be obtained from the parabolic subalgebras of su(2, 2) and their refined Langland decompositions, by using Theorems 2.2 and 2.3. The four standard parabolic subalgebras are parametrized by the subsets , ∅,

1 = { f1 − f2 } and 2 = { 2 f2 } of . The case = We have e = a = n = {0}, hence the refined Langlands decomposition is p = su(2, 2) + {0} + {0} + {0}. Thus, for a connected closed ˆ of P , it is only possible to have G ˆ = L = L = co-compact subgroup G SU(2, 2). Then the only transitive action on A SU(2,2) coming from = is that of the full isometry group SU(2, 2), and we have the description of A SU(2,2) as the symmetric space SU(2, 2)/S(U(2) × U(2)). The associated reductive decomposition is the Cartan decomposition su(2, 2) = s(u(2) ⊕ u(2)) + p, and the corresponding homogeneous quaternionic Kähler structure is S = 0. The case = ∅ In this case, l = l ∅ + e ∅ + a∅ is commutative, l ∅ = {0}, e ∅ = Z k (a), and a∅ = a. Hence the refined Langlands decomposition of the minimal parabolic subalgebra is p∅ = {0} + Z k (a) + a + n. For each connected closed subgroup E of E∅ A ∼ = U(1) × R2 we get a co-compact subgroup EN of SU(2, 2), where N is the nilpotent factor in the Iwasawa decomposition of SU(2, 2). In order to obtain a transitive action it is sufficient that the projection of the Lie algebra e ⊂ Z k (a) + a of E to a be surjective. For each choice of a two-dimensional subspace e of Z k (a) + a =  C, A1 , A2  such as C ∈ e, we obtain a Lie subalgebra e + n of su(2, 2) which generates a ˆ = EN which acts simply transitively on A SU(2,2) . connected closed subgroup G We can suppose that e = eλ,μ is generated by two elements of the form λC + A1 , μC + A2 , with λ, μ ∈ R. This gives the reductive decomposition gˆ λ,μ = {0} + gˆ λ,μ associated to the description A SU(2,2) = Eλ,μ N, where gˆ λ,μ =  λC + A1 , μC + A2 , U 1 , U 2 , X1 , X1 , X2 , X2 . Distinct values of λ, μ define non-isomorphic Lie algebras. If λ = μ = 0, we have the choice e = a, which gives the usual description of A SU(2,2) as the solvable Lie group AN. We get a two-parameter family of structures Sλ,μ associated to the family of reductive decompositions gˆ λ,μ = {0} + gˆ λ,μ . With the identifications in (2.2), where now m = gˆ λ,μ = eλ,μ + n, and by using (2.4), we obtain that S = Sλ,μ is given at o by Table 1, with δ = ε = 1.

X2 −X2 −ε X1 ε X1

X2

−X2

ε X1

−ε X1

−X1

−X1

ε X2

ε X2

0

−X1

−X1

−ε X2

−ε X2

S X1

S X1

S X2

S X2

0 2δ A2

2A1

0

0

−2δU 2

−2U 1

SU2

0 0

SU1

0 0

U2

0 0

U1

0 0

A2

S A1 S A2

A1

Table 1 Homogeneous structures at o for SU(2, 2)/S(U(2) × U(2))

2ε(U 1 − U 2 )

0

0

2(A1 + A2 )

2λX1 2μX1 X2 δ X2

X1

0

−2ε(U 1 − U 2 )

2(A1 + A2 )

0

−δ X2

−X2

−2λX1 −2μX1

X1

0

2ε(A1 − A2 )

2(U 1 + U 2 )

0

2λX2 2μX2 X1 δ X1

X2

2ε(A1 − A2 )

0

0

−2(U 1 + U 2 )

−δ X1

−X1

−2λX2 −2μX2

X2

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 = ∇ − Sλ,μ is Each structure Sλ,μ is also characterized by the fact that ∇ the canonical connection for the Lie group Eλ,μ N, which is the connection for which every left-invariant vector field on Eλ,μ N is parallel. The choice e = Z k (a) + a gives A SU(2,2) ≡ (U(1) × R2 )N/U(1), and the associated reductive decomposition is gˆ = h + m, where gˆ = p∅ , h = Z k (a) and m = a + n. The corresponding structure coincides with the above structure Sλ,μ , for λ = μ = 0. The case = 1 Then [ 1 ] = { ±( f1 − f2 ) }, and one has l 1 =  A2 − A1  + Z k (a) +  X2 , Y2 , X2 , Y2 , e 1 = {0}, a 1 =  A1 + A2 , and n 1 = g f1 + f2 + g2 f1 + g2 f2 =  X1 , X1 , U 1 , U 2 . Hence, the corresponding parabolic subalgebra is p 1 = l 1 + {0} + a 1 + n 1 , with ⎧⎛ ⎫ ⎞ ir z w s ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ − z ¯ −ir −s w ¯ ∼ ⎜ ⎟ l 1 = ⎝ : r, s ∈ R , z, w ∈ C = sl(2, C). w¯ −s −ir −z¯ ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ s w z ir The connected subgroup E = A 1 ∼ = R of P 1 with Lie algebra a 1 is the ˆ = LEN 1 of SU(2, 2), only possible choice to get a co-compact subgroup G ∼ with L = L 1 = Sl(2, C), which acts transitively on A SU(2,2) . The isotropy algebra is h = gˆ ∩ k = l 1 ∩ s(u(2) ⊕ u(2)) =  C, (X2 )k , (X2 )k  ∼ = su(2), and we get the reductive decomposition gˆ = h + m, where gˆ = p 1 , and m =  A1 , A2 , U 1 , U 2 , X1 , X1 , (X2 )p , (X2 )p , which is associated to the description A SU(2,2) ≡ Sl(2, C)R N 1 /SU(2). By using (2.4), we obtain that the corresponding structure S is given at o by Table 1, with λ = μ = ε = 0, δ = 1. The case = 2 Then [ 2 ] = { ±2 f2 }, and we have l 2 =  A2 , U 2 , V2 , e 2 = Z k (a), a 2 =  A1 , n 2 = g f1 + f2 + g f1 − f2 + g2 f1 =  X1 , X1 , X2 , X2 , U 1 . The corresponding parabolic subalgebra is p 2 = l 2 + Z k (a) + a 2 + n 2 , with ⎧⎛ ⎫ ⎞ ix 0 0 z ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ 0 0 0 0 ∼ ⎜ ⎟ l 2 = ⎝ : x ∈ R , z ∈ C = su(1, 1) ∼ = sl(2, R). 0 00 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ z¯ 0 0 −ix For each connected closed subgroup E of P 2 whose Lie algebra e is a non-trivial subspace of e 2 + a 2 , e = e 2 , we get a co-compact subgroup ∼ ˆ = LEN 2 of SU(2, 2), with L = L ∼ G

2 = SU(1, 1) = Sl(2, R), which acts transitively on A SU(2,2) . If dim E = 1, we get the description A SU(2,2) ≡ SU(1, 1)R N 2 /U(1). In fact, if e is an one-dimensional subspace of e 2 + a 2 , with e = e 2 , then e = eλ is spanned by an element of the form λC + A1 , for λ ∈ R, so one gets a oneparameter family of reductive decompositions gˆ λ = h + mλ , where h = gˆ ∩ k = l 2 ∩ s(u(2) ⊕ u(2)) =  (U 2 )k  ∼ = u(1) is the isotropy algebra and mλ =  λC + A1 , A2 , U 1 , (U 2 )p , X1 , X1 , X2 , X2 . By (2.4), we have that the corresponding

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one-parameter family of structures Sλ is given at o by Table 1, with μ = δ = 0, ε = 1. In particular, λ = 0 corresponds to the choice e = a 2 . If dim E = 2, then E ∼ = U(1) × R, so A SU(2,2) ≡ SU(1, 1)(U(1) × R)N 2 / (U(1) × U(1)). The natural associated reductive decomposition is gˆ = h + m , where gˆ = p 2 , h =  C, (U 2 )k  ∼ = u(1) ⊕ u(1), m =  A1 , A2 , U 1 , (U 2 )p , X1 , X1 , X2 , X2 , and the corresponding structure S coincides with the previous Sλ , for λ = 0. In particular, we have proved Proposition 3.1 The space A SU(2,2) admits the homogeneous quaternionic Kähler structures S (associated to the corresponding homogeneous description ˆ G/H) given by S 0 Table 1 (λ, μ ∈ R; δ = ε = 1) Table 1 (λ = μ = ε = 0, δ = 1) Table 1 (λ ∈ R; μ = δ = 0, ε = 1)

ˆ G/H SU(2, 2)/S(U(2) × U(2)) Eλ,μ N (Eλ,μ ∼ = R2 ) Sl(2, C)R N 1 /SU(2) SU(1, 1)R N 2 /U(1)

where N is the nilpotent factor in the Iwasawa decomposition SU(2, 2) = S(U(2) × U(2))AN and 1 and 2 are the subsets of simple restricted roots defining the intermediate parabolic subalgebras of (su(2, 2), a). 3.2 The Quaternionic Hyperbolic Space A Sp(2,1) = HH(2) 3.2.1 The Quaternion-Hermitian Structure of a + n ⊂ sp(2, 1) The Lie algebra sp(2, 1) can be described as the subalgebra of gl(6, C) of all ma¯ T J2,1 + J2,1 X = 0, where J2,1 = diag(−1, −1, trices X ∈ sp(3, C) satisfying X ¯ T defines the 1, −1, −1, 1). The involution τ of sp(2, 1) given by τ (X) = − X Cartan decomposition sp(2, 1) = k + p, where ⎧⎛ ⎫ ⎞ ia z 0 u w 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ −z¯ ib 0 w ⎟ ⎪ v 0 ⎪ ⎪ ⎪ ⎟ ⎪ ⎨⎜ ⎬ ⎜ 0 0 ic 0 ⎟ 0 α a, b , c ∈ R , ∼ ⎟ : k= ⎜ = sp(2) ⊕ sp(1), ⎜ −u¯ −w¯ 0 −ia z¯ 0 ⎟ ⎪ ⎪ ⎜ ⎟ z, u, v, w, α ∈ C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎝ −w¯ −v¯ 0 −z −ib 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 −α¯ 0 0 −ic ⎧⎛ 0 ⎪ ⎪ ⎪ ⎪ ⎜ 0 ⎪ ⎪ ⎨⎜ ⎜ p¯ 1 p= ⎜ ⎜ 0 ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ 0 ⎪ ⎪ ⎩ q¯ 1

0 0 p¯ 2 0 0 q¯ 2

⎫ ⎞ p1 0 0 q1 ⎪ ⎪ ⎪ ⎪ p2 0 0 q2 ⎟ ⎪ ⎟ ⎪ ⎬ ⎟ 0 q1 q2 0⎟ , p , q , q ∈ C : p . 1 2 1 2 ⎟ q¯ 1 0 0 − p¯ 1 ⎟ ⎪ ⎪ ⎪ ⎪ q¯ 2 0 0 − p¯ 2 ⎠ ⎪ ⎪ ⎭ 0 − p1 − p2 0

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The element A0 of p obtained by taking p1 = 1 and p2 = q1 = q2 = 0 generates a maximal R-diagonalizable subalgebra a of sp(2, 1). The set of roots corresponding to a is  = { ± f0 , ±2 f0 }, where f0 ∈ a∗ is given by f0 (A0 ) = 1. The set = { f0 } is a system of simple roots and the corresponding positive root system is  + = { f0 , 2 f0 }. The positive root vector spaces are

g f0

g2 f0

⎫ ⎧⎛ ⎞ 0 z 0 0 w 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ −z¯ 0 z¯ w 0 w ⎟ ⎪ ⎪ ⎪ ⎪ ⎟ ⎬ ⎨⎜ ⎜ 0 z 0 0 w 0⎟ ⎟ : z, w ∈ C , = ⎜ ⎜ 0 −w¯ 0 0 z¯ 0 ⎟ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ⎪ ⎝ −w¯ ⎠ 0 w ¯ −z 0 −z ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 w¯ 0 0 −z¯ 0

⎧⎛ ia ⎪ ⎪ ⎪ ⎪ ⎜ 0 ⎪ ⎪ ⎨⎜ ⎜ ia = ⎜ ⎜ −u¯ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ 0 ⎪ ⎪ ⎩ u¯

⎫ ⎞ 0 − ia u 0 u ⎪ ⎪ ⎪ ⎪ 0 0 0 0 0⎟ ⎪ ⎟ ⎪ ⎬ 0 − ia u 0 u⎟ ⎟ : a ∈ R, u ∈ C . 0 u¯ − ia 0 − ia ⎟ ⎪ ⎟ ⎪ ⎪ ⎪ 0 0 0 0 0⎠ ⎪ ⎪ ⎭ 0 − u¯ ia 0 ia

(3.3)

(3.4)

The root vector spaces for the respective opposite roots are the corresponding sets of opposite conjugate transpose matrices. Now, let X1 , U 1 and U 1 be the elements of g2 f0 in (3.4) obtained by putting (a, u) = (−1, 0), (0, 1) and (0, i), respectively, and denote by Y1 , V1 and V1 the corresponding respective elements of g−2 f0 . On the other hand, let X2 , X2 , U 2 and U 2 be the elements of g f0 in (3.3) obtained by setting (z, w) = (1, 0), (−i, 0), (0, 1) and (0, i), respectively, and let Y2 , Y2 , V2 and V2 denote the corresponding respective elements of g− f0 . Then, g f0 =  X2 , X2 , U 2 , U 2 , g2 f0 =  X1 , U 1 , U 1 , g− f0 =  Y2 , Y2 , V2 , V2 , g−2 f0 =  Y1 , V1 , V1 , and we have the Iwasawa decomposition sp(2, 1) = k + a + n, where n = g f0 + g2 f0 =  X1 , U 1 , U 1 , X2 , X2 , U 2 , U 2 .

The centralizer of a in k is ⎧⎛ ⎫ ⎞ ia 0 0 u 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ 0 ib 0 0 v 0 ⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎨⎜ ⎬ ⎟ 0 0 ia 0 0 −u ⎟ a, b ∈ R, ∼ ⎜ Z k (a) = ⎜ : = sp(1) ⊕ sp(1), (3.5) 0 0⎟ ⎪ ⎪ ⎪ ⎜ −u¯ 0 0 −ia ⎟ u, v ∈ C ⎪ ⎪ ⎪ ⎪ ⎪ ⎝ 0 −v¯ 0 0 −ib 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 u¯ 0 0 −ia

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and it is generated by the elements C1 , C2 , C3 , D1 , D2 , D3 in (3.5) obtained by taking (a, b , u, v) = (1, 0, 0, 0), (0, 0, 1, 0), (0, 0, i, 0), (0, 1, 0, 0), (0, 0, 0, 1) and (0, 0, 0, i), respectively. The subspaces  C1 , C2 , C3  and  D1 , D2 , D3  are ideals of Z k (a) isomorphic to sp(1). The elements E1 , E2 , E3 of k = sp(2) ⊕ sp(1) ⊂ sp(2, 1) given by ⎛

⎞ 00000 0 ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎜0 0 i 0 0 0⎟ ⎜ ⎟ ⎜0 0 0 0 0 0⎟, ⎜ ⎟ ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎝0 0 0 0 0 0⎠ 0 0 0 0 0 −i

⎛

⎞ 00 0000 ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎜0 0 0 0 0 1⎟ ⎜ ⎟ ⎜0 0 0 0 0 0⎟, ⎜ ⎟ ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎝0 0 0 0 0 0⎠ 0 0 −1 0 0 0

⎛

⎞ 000000 ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎜0 0 0 0 0 i⎟ ⎜ ⎟ ⎜0 0 0 0 0 0⎟, ⎜ ⎟ ⎜0 0 0 0 0 0⎟ ⎜ ⎟ ⎝0 0 0 0 0 0⎠ 00 i000

respectively, satisfy [E1 , E2 ] = 2E3 , [E2 , E3 ] = 2E1 , [E3 , E1 ] = 2E2 , and generate a compact ideal u ∼ = sp(1) of k. The isotropy representation u → gl(p) defines a quaternionic structure on A Sp(2,1) . The isomorphisms p ∼ = sp(2, 1)/ k ∼ = a + n allow to obtain the complex structures Ja (a = 1, 2, 3), acting on a + n. The action on the elements of the basis { A0 , X1 , U 1 , U 1 , X2 , X2 , U 2 , U 2 } of a + n is given in the following table. A0

X1

U1

U 1

X2

X2

U2

U 2

J1 −X1 A0 U 1 −U 1 −X2 X2 U 2 −U 2 J2 −U 1 −U 1 A0 X1 −U 2 −U 2 X2 X2 J3 −U 1 U 1 −X1 A0 −U 2 U 2 −X2 X2 The above basis of a + n is orthonormal with respect to the scalar product  ,  induced in a + n by the isomorphism p ∼ = a + n and 1/32B| p × p , where B is the Killing form of sp(2, 1), and (a + n,  , , J1 , J2 , J3 ) is a quaternion-Hermitian vector space. 3.2.2 Homogeneous Descriptions and Structures of A Sp(2,1) We now determine the homogeneous descriptions and structures of A Sp(2,1) . The two only parabolic subalgebras of sp(2, 1) are parametrized by the subsets and ∅ of = { f0 }. We will use them and their refined Langland decompositions to obtain, by means of Theorems 2.2 and 2.3, the homogeneous descriptions of A Sp(2,1) = HH(2). The case = We have e = a = n = {0}, hence the refined Langlands decomposition is p = sp(2, 1) + {0} + {0} + {0}. Theorem 2.2 then says that ˆ = Sp(2, 1). Thus the only connected closed co-compact subgroup of P is G

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the only transitive action coming from = is that of the full isometry group Sp(2, 1). This gives the description of A Sp(2,1) = HH(2) as the symmetric space Sp(2, 1)/(Sp(2) × Sp(1)). The associated reductive decomposition is the Cartan decomposition sp(2, 1) = (sp(2) ⊕ sp(1)) + p, and the corresponding structure is S = 0. The case = ∅ We have l = a + Z k (a) = l ∅ + e ∅ + a∅ , with l ∅ = {0}, e ∅ = Z k (a), a∅ = a, so the refined Langlands decomposition of the corresponding parabolic subalgebra is p∅ = {0} + Z k (a) + a + n = {0} + (sp(1) ⊕ sp(1)) + a + (g f0 + g2 f0 ). For each connected closed subgroup E of E∅ A ∼ = Sp(1)Sp(1)R we get a co-compact subgroup EN of Sp(2, 1). In order to get a transitive action on A Sp(2,1) it is sufficient, by Theorem 2.3, that the projection of the Lie algebra e ⊂ Z k (a) + a to a be surjective. Now, for each one-dimensional subspace e ⊂ Z k (a) + a =  C j, D j, A0  j=1,2,3 such that the projection of e to a is an isomorphism, we get a Lie subalgebra e + n of sp(2, 1) which generˆ = EN acting simply transitively on ates a connected closed Lie subgroup G A Sp(2,1) . We can suppose that e = eλ,μ is generated by one element of the  form Aˆ 0 = A0 + 3j=1 (λ jC j + μ j D j), with λ = (λ1 , λ2 , λ3 ), μ = (μ1 , μ2 , μ3 ) ∈ R3 . This gives the reductive decomposition gˆ λ,μ = {0} + gˆ λ,μ associated to the description A Sp(2,1) = Eλ,μ N, where gˆ λ,μ =  Aˆ 0 , X1 , U 1 , U 1 , X2 , X2 , U 2 , U 2 . If λ = μ = (0, 0, 0), we have e = a, which gives the usual description of A Sp(2,1) as the solvable Lie group AN. Then we have a six-parameter family of structures S = Sλ,μ associated to the reductive decompositions gˆ λ,μ = {0} + gˆ λ,μ . For each λ, μ ∈ R3 , and with the identifications in (2.2), where now m = gˆ λ,μ = eλ,μ + n, we apply (2.4) to obtain the homogeneous quaternionic Kähler structure S = Sλ,μ , which is given at o by Table 2. If dim e > 1 we obtain other subgroups E of E∅ A ∼ = Sp(1)Sp(1)R such that EN acts transitively on A Sp(2,1) . Such a E is isomorphic to some group of the form U(1)R, U(1)U(1)R, Sp(1)R, Sp(1)U(1)R or Sp(1)Sp(1)R. However, the natural reductive decompositions defined by their actions do not provide new homogeneous Riemannian structures. We have proved Proposition 3.2 The space A Sp(2,1) admits the homogeneous quaternionic Kähler structures S (associated to each corresponding homogeneous description ˆ G/H) given by

S 0 Table 2

ˆ G/H Sp(2, 1)/(Sp(2) × Sp(1)) (λ, μ ∈ R3 ) Eλ,μ N (Eλ,μ ∼ = R)

where N is the nilpotent factor in the Iwasawa decomposition of Sp(2, 1).

−X2

X2

U 2

−X2

−X2

−U 2

−U 2

S X2

S X2

SU2

U 2

−U 2

−U 2

0

−2U 1

SU1

S

0

0

−2U 1

SU1

X2

X2

−U 2

2A0

0

2A0

−2X1

S X1

2λ1 U 1 +2λ3 X1

2λ2 U 1 −2λ3 U 1

0

U1

X1

S A0

A0

Table 2 Homogeneous structures at o for H(2)

X2

−X2

U2

−U 2

2A0

0

0

−2λ1 U 1 −2λ2 X1

U 1

−U 1

−U 1

−X1

A0

−U 2

−U 2

−U 1

U 1

A0

X1

U2

−U 2

X2

−(λ3 + μ3 )U 2

+(λ3 − μ3 )U 2 −X2

(λ1 − μ1 )X2 +(λ2 + μ2 )U 2

X2

(μ1 − λ1 )X2 +(λ2 − μ2 )U 2

X2

X1

A0

−U 1

U1

−X2

X2

U 2

+(λ3 + μ3 )X2

(λ1 + μ1 )U 2 +(μ2 − λ2 )X2

U2

A0

−X1

U1

U 1

X2

X2

−U 2

+(μ3 − λ3 )X2

−(λ1 + μ1 )U 2 −(λ2 − μ2 )X2

U 2

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3.3 The Exceptional Space AG2(2) = G2(2) /SO(4) 3.3.1 The Quaternion-Hermitian Structure of a + n ⊂ g2(2) The non-compact real form g2(2) of the exceptional Lie algebra g2 can be realized as a subalgebra of so(4, 3). Each element X ∈ g2(2) can be represented by a real matrix of the form ⎛ ⎞ 0 x1 + u1 −x2 − u2 2u3 s1 y1 + v1 y2 − v2 ⎜ −x1 − u1 0 x3 + u3 2u2 y1 − v1 s2 y3 + v3 ⎟ ⎜ ⎟ ⎜ x2 + u2 −x3 − u3 ⎟ 0 2u y + v y − v s3 1 2 2 3 3 ⎜ ⎟ ⎜ −2u3 ⎟ , (3.6) −2u −2u 0 2v 2v 2v 2 1 3 2 1 ⎜ ⎟ ⎜ ⎟ s y − v y + v 2v 0 x − u −x + u 1 1 1 2 2 3 1 1 2 2⎟ ⎜ ⎝ y1 + v1 s2 y3 − v3 2v2 −x1 + u1 0 x 3 − u3 ⎠ y2 − v2 y3 + v3 s3 2v1 x2 − u2 −x3 + u3 0 with s1 + s2 + s3 = 0. The involution τ of g2(2) given by τ (X) = −X T defines the Cartan decomposition g2(2) = k + p, where k = (so(4) ⊕ so(3)) ∩ g2(2) ∼ = so(4). The subspace a of p defined by ⎧⎛ ⎫ ⎞ 0 0 0 0 s1 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ 0 0 0 0 0 s2 0 ⎟ ⎪ ⎪ ⎪ ⎟ ⎪ ⎜ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎨⎜ 0 0 0 0 0 0 s3 ⎟ ⎬ ⎟ ⎜ a = ⎜ 0 0 0 0 0 0 0 ⎟ : s1 + s2 + s3 = 0 ⎪ ⎪ ⎟ ⎪ ⎜ ⎪ ⎪ ⎪ ⎪⎜ s1 0 0 0 0 0 0 ⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎠ ⎝ 0 0 0 0 0 0 s ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ 0 0 s3 0 0 0 0 is a maximal R-diagonalizable subalgebra of g2(2) , and Z k (a) = {0}. The elements A1 and A2 of a defined by (s1 , s2 , s3 ) = (1, −1, 0) and (−2, 1, 1), respectively, generate a. Let f j be the member of a∗ whose value on the above matrix of a is s j. Then f1 + f2 + f3 = 0. We consider the (weak) order in a∗ defined by choosing the element A0 ∈ a given by (s1 , s2 , s3 ) = (−1, −2, 3) and by saying that f ∈ a∗ is positive if f (A0 ) > 0. Then, the sets of positive roots and simple roots of (g2(2) , a) are  + = { f1 − f2 , f3 − f1 , f3 − f2 , f3 , − f2 , − f1 } and = { f1 − f2 , − f1 }, respectively. Now, we will give the generators X j, Y j, U j, V j, 1  j  3, of the root spaces g f , f ∈ . The matrix X in (3.6) represents X1 if x1 = y1 = 1/2,

X2 if x2 = y2 = 1/2,

X3 if x3 = −y3 = −1/2,

Y1 if x1 = −y1 = −1/2, Y2 if x2 = −y2 = −1/2, Y3 if x3 = y3 = −1/2, U 1 if u1 = v1 = −1/2,

U 2 if u2 = −v2 = 1/2,

U 3 if u3 = −v3 = −1/2,

V1 if u1 = −v1 = 1/2,

V2 if u2 = v2 = −1/2,

V3 if u3 = v3 = 1/2,

with all the other values x j, y j, u j, v j, s j = 0 for each of the 12 cases. We have g f1 − f2 =  X1 , g− f1 + f2 =  Y1 , g f3 − f1 =  X2 , g− f3 + f1 =  Y2 , g f3 − f2 =  X3 ,

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g− f3 + f2 =  Y3 , g f3 =  U 1 , g− f3 =  V1 , g− f2 =  U 2 , g f2 =  V2 , g− f1 =  U 3 , g f1 =  V3 . Then g2(2) = a + f ∈ g f , and we have the Iwasawa decomposition g2(2) = k + a + n, where n = f ∈ + g f =  X1 , X2 , X3 , U 1 , U 2 , U 3 . The elements E1 , E2 , E3 of k = so(4) ⊂ g2(2) , which are represented by the matrices ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 01 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 0 ⎜ −1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 1 0 0 0 ⎟ ⎜ 1 0 0 0 0 0 0 ⎟ ⎜ 0 −1 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 −1 0 0 0 0 ⎟ , ⎜ 0 −1 0 0 0 0 0 ⎟ , ⎜ −1 0 0 0 0 0 0 ⎟ , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 0⎟ ⎜0 0 0 0 0 0 0⎟ ⎜ 0 0 0 0 0 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 0 0⎠ ⎝0 0 0 0 0 0 0⎠ ⎝ 0 0 0 0 0 0 0⎠ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

respectively, satisfy [E1 , E2 ] = 2E3 , [E2 , E3 ] = 2E1 , [E3 , E1 ] = 2E2 , and generate a compact ideal u ∼ = sp(1) of k, such that the isotropy representation u → gl(p) defines a quaternionic structure on AG2(2) . Through the isomorphisms p ∼ = g2(2) / k ∼ = a + n, the action of each Ea on p defines the complex structure Ja (a = 1, 2, 3), acting on the elements A1 , A2 , X j, U j (1  j  3) of a + n as follows.

J2

A1 −2X1 X2−U 2

A2 3X1+U 1 −3X2+U 2

J3

−X3−U 3

2U 3

J1

X1 1/2A1 −1/2(X3+U 3 ) 1/2(−X2+U 2 )

X2 1/2(X3−U 3 ) 1/2(A1+A2 )

X3 −1/2(X2+U 2 ) 1/2(X1−U 1 )

U1 −3/2A1−A2 1/2(3X3−U 3 )

U2 1/2(3X3+U 3 ) 1/2(3A1+A2 )

U3 1/2(3X2−U 2 ) 1/2(3X1+U 1 )

1/2(X1+U 1 )

A1+1/2A2

−1/2(3X2+U 2 )

1/2(−3X1+U 1 )

−1/2A2

We consider the scalar product  ,  induced in a + n by the isomorphisms p∼ = a+n and (1/16)B| p×p , where B is the Killing form of g2(2) . Then, A1 ,A1  = 1, A2 , A2  = 3, A1 , A2  = −3/2, X j, X j = 1/4, U j, U j = 3/4, and (a + n,  , , J1 , J2 , J3 ) is a quaternion-Hermitian vector space. 3.3.2 Homogeneous Descriptions and Structures of AG2(2) The four standard parabolic subalgebras of g2(2) are parametrized by the subsets , ∅, 1 = { f1 − f2 } and 2 = { − f1 } of . By using Theorems 2.2 and 2.3 we will obtain the homogeneous descriptions of AG2(2) . The case = We have [ ] = , e = a = n = {0}, and the refined Langlands decomposition is p = g2(2) + {0} + {0} + {0}. Here one only gets the transitive action of the full isometry group G2(2) on AG2(2) , and we have the description of AG2(2) as the symmetric space G2(2) /SO(4). The associated reductive decomposition is the Cartan decomposition g2(2) = k + p, with k ∼ = so(4), and the corresponding structure is S = 0. The case = ∅ In this case, l ∅ = {0}, e ∅ = Z k (a) = {0}, and a∅ = a, and hence the refined Langlands decomposition of the minimal parabolic subalgebra is ˆ p∅ = {0} + {0} + a + n. In this case, the only connected co-compact subgroup G

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which acts transitively on AG2(2) is provided by E = A, so we have the descripˆ = AN, where N is the nilpotent tion of AG2(2) as the solvable Lie group G factor in the Iwasawa decomposition of G2(2) . The associated reductive decomposition is a + n = {0} + (a + n) and, having in mind that in this case m = a + n, by using (2.4) we obtain that the corresponding homogeneous quaternionic Kähler structure S is given by Table 3 with δ = ε = 1. The case = 1 Then [ 1 ] = { ±( f1 − f2 ) }, and p 1 = l 1 + {0} + a 1 + n 1 ,  where a 1 =  3A1 + 2A2 , n 1 = f ∈ + , f = f1 − f2 g f =  X2 , X3 , U 1 , U 2 , U 3 , and

l 1

⎧⎛ 0 x 0 ⎪ ⎪ ⎪ ⎪ ⎜ −x 0 0 ⎪ ⎪ ⎪⎜ ⎪ ⎨⎜ ⎜ 0 0 0 = ⎜ ⎜ 0 0 0 ⎪ ⎪ ⎜ s y 0 ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ y −s 0 ⎪ ⎪ ⎩ 0 0 0

⎞⎫ 0 s y 0 ⎪ ⎪ ⎪ ⎪ 0 y −s 0 ⎟ ⎟⎪ ⎪ ⎪ ⎟ 0 0 0 0 ⎟⎪ ⎬ ⎟ 0 0 0 0 ⎟ =  A1 , X1 , Y1  ∼ = sl(2, R). ⎪ ⎪ 0 0 x 0⎟ ⎪ ⎟⎪ ⎪ 0 −x 0 0 ⎠⎪ ⎪ ⎪ ⎭ 0 0 0 0

The connected Lie subgroup E = A 1 ∼ = R of P 1 with Lie algebra a 1 is the ˆ = LEN 1 of G2(2) , with only possible choice to get a co-compact subgroup G ∼ ˆ L = L 1 = Sl(2, R), and G acts transitively on AG2(2) . The isotropy algebra is h = gˆ ∩ k = l 1 ∩ k =  (X1 )k  ∼ = so(2). We have the reductive decomposition gˆ = h + m, where gˆ = p 1 , and m =  A1 , A2 , (X1 )p , X2 , X3 , U 1 , U 2 , U 3 , which is associated to the description AG2(2) ≡ Sl(2, R)R N 1 /SO(2). By using (2.4), we obtain that the corresponding homogeneous quaternionic Kähler structure S is given at o by Table 3 with δ = 0, ε = 1. The case = 2 Then [

2 ] = {± f1 } and p 2 = l 2 + {0} + a 2 + n 2 , where a 2 =  2A1 + A2 , n 2 = f ∈ + , f =− f1 g f =  X1 , X2 , X3 , U 1 , U 2 , and

l 2

⎧⎛ 0 0 0 ⎪ ⎪ ⎪⎜ ⎪ 0 0 u ⎪ ⎪ ⎜ ⎪ ⎪ 0 −u 0 ⎨⎜ ⎜ −2u 0 0 = ⎜ ⎜ ⎪ ⎪ ⎜ −2s 0 0 ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ 0 s −v ⎪ ⎪ ⎩ 0 v s

⎞⎫ 2u −2s 0 0 ⎪ ⎪ ⎪ ⎪ 0 0 s v⎟ ⎟⎪ ⎪ ⎪ ⎟ 0 0 −v s ⎟⎪ ⎬ ⎟ 0 2v 0 0 ⎟ =  A2 , U 3 , V3  ∼ = sl(2, R). ⎪ ⎪ 2v 0 0 0 ⎟ ⎪ ⎟⎪ ⎪ 0 0 0 −u ⎠⎪ ⎪ ⎪ ⎭ 0 0 u 0

In this case we also have only one possible choice for E as the connected subgroup A 2 ∼ = R of P 2 whose Lie algebra is a 2 . We thus get a co-compact ˆ = LEN 2 of G2(2) , with L = L ∼ subgroup G

2 = Sl(2, R), which acts transitively on AG2(2) , so AG2(2) ≡ Sl(2, R)R N 2 /SO(2), whose natural associated

1/2δ A1

−1/2X3

−1/2X2

0

3δ X1

−3X2

0

−U 1

U2

−2εU 3

−δ2X1

X2

−X3

0

−U 2

εU 3

S X1

S X2

S X3

SU1

SU2

SU3

−1/2εU 2

−1/2U 3

0

0

0

X1

0

S A2

A2

0

0

S A1

A1

Table 3 Homogeneous structures at o for G2(2) /SO(4) X2

−1/2εU 1

0

1/2U 3

1/2X1

1/2A1 + 1/2A2

1/2δ X3

0

0

X3

0

−1/2U 1

1/2U 2

A1 + 1/2A2

1/2X1

−1/2δ X2

0

0

ε(3/2X2 + U 2 )

3/2X3 − U 3

3/2A1 + A2

1/2U 2

1/2U 3

0

0

0

U1

ε(3/2X1 − U 1 )

3/2A1 + 1/2A2

−3/2X3 − U 3

−1/2U 1

0

−1/2δU 3

0

0

U2

1/2ε A2

3/2X1 + U 1

−3/2X2 + U 2

0

−1/2U 1

1/2δU 2

0

0

U3

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reductive decomposition is gˆ = h + m, where gˆ = p 2 , h = gˆ ∩ k = l 2 ∩ k =  (U 3 )k  ∼ = so(2) and m =  A1 , A2 , X1 , X2 , X3 , U 1 , U 2 , (U 3 )p . By (2.4), we have that the corresponding structure S is given at o by Table 3 with δ = 1, ε = 0. We have Proposition 3.3 The space AG2(2) admits the homogeneous quaternionic Kähler ˆ structures S (associated to each corresponding homogeneous description G/H) given by S 0 Table 3, δ = ε = 1 Table 3, δ = 0, ε = 1 Table 3, δ = 1, ε = 0

ˆ G/H G2(2) /SO(4) AN (A ∼ = R2 ) Sl(2, R)R N 1 /SO(2) Sl(2, R)R N 2 /SO(2)

where N is the nilpotent factor in the Iwasawa decomposition G2(2) = SO(4)AN and 1 and 2 are the subsets of simple restricted roots defining the intermediate parabolic subalgebras of (g2(2) , a). 3.4 Types of Homogeneous Quaternionic Kähler Structures For each parabolic subgroup P of the full connected isometry group G of each non-compact quaternion-Kähler symmetric space M of dimension 8, the ˆ of G acting transitively on M are of the form G ˆ = LEN as in subgroups G Theorem 2.3, where L = L . Moreover, E is a connected closed subgroup of E

A such that the projection of its Lie algebra e ⊂ e + a to a is surjective. If this projection is an isomorphism we say that E is minimal. Then we have Theorem 3.4 Let G = K AN be the Iwasawa decomposition of each of the groups SU(2, 2), Sp(2, 1), and G2(2) . The homogeneous descriptions of each eight-dimensional non-compact quaternion-Kähler symmetric space are of the ˆ ˆ = L EN , with L either non-compact simple or trivial form G/H, where G



and N nilpotent. If E is minimal, then it is simply-connected and abelian, and in this case the corresponding types of homogeneous quaternionic Kähler structures are given in Table 4 (where the figure on the fifth column, if any, stands for the number of parameters of the corresponding n-parametric family of homogeneous quaternionic Kähler structures).

Proof In order to determine the types of non-zero homogeneous quaternionic Kähler structures (see Theorem 1.3) on the three spaces, we obtain for each structure the forms θ i satisfying Eq. 1.3, by using the formula (2.6).

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Table 4 Homogeneous descriptions and types of structures for the minimal case G/K



L EN /H

dim E

A SU(2,2)

∅

1

2 ∅ ∅ ∅ ∅

1

2

SU(2, 2)/S(U(2) × U(2)) Eλ,μ N (λ, μ ∈ R) Sl(2, C)A 1 N 1 /SU(2) SU(1, 1)Eλ N 2 /U(1) Sp(2, 1)/(Sp(2) × Sp(1)) Eλ,μ N (λ, μ ∈ R3 \ {0}) E0,μ N (μ ∈ R3 \ {0}) AN = E0,0 N G2(2) /SO(4) AN Sl(2, R)A 1 N 1 /SO(2) Sl(2, R)A 2 N 2 /SO(2)

0 2 1 1 0 1 1 1 0 2 1 1

A Sp(2,1)

AG2(2)

n 2 1 6 3

type {0} QK12345 QK135 QK12345 {0} QK12345 QK1345 QK134 {0} QK12345 QK12345 QK12345

Case 1 The space A SU(2,2) . The values of the forms θ i at o ∈ A SU(2,2) corresponding to the homogeneous quaternionic Kähler structure S given in Table 1 are as follows

θ1 θ2 θ3

X1

X2

X2

−δ 0 0 0 −2 0 0 0 −2

0 2ε 0

0 0 2ε

A1

A2 U 1 U 2

2λ 0 0

2μ 0 0

1 0 0

X1

If = ∅, (δ = ε = 1), the structure S = Sλ,μ  (λ, μ ∈ R) decomposes 3 as S =  + T, with  ∈ QK12 (i.e.,  X Y = (1/2) a=1 θ a (X)Ja Y, see [6, §§3.1]) and T ∈ QK345 . The condition for the tensor  to be in QK1 is θ; and this condition is not satisθ a = θ ◦ Ja , a = 1, 2, 3, for some 3 1-form a fied. In turn,  ∈ QK2 if θ ◦ J = 0, which is not satisfied for insa a=1 tance by A1 . Hence  ∈ QK12 \ (QK1 ∪ QK2 ). The relevant trace of T is c12 (T) = 7A1/2 + 3A2/2 + λU 1 − μU 2 , ·. On the other hand, since dim M = 8, the form defining the QK3 - component of T is given by ϑ = c12 /10. Hence 20ϑ = 7A1 + 3A2 + 2λU 1 − 2μU 2 , · , so the QK3 -component of T ˆ ˆ does not vanish either. Consider 3 now the operator F : V → V defined by F(T) XY Z = TY Z X −T Z Y X + a=1 (T Ja Y Ja Z X − T Ja Z Ja Y X ), with eigenvalues 2 and −4 and respective eigenspaces QK34 and QK5 (see Theorem 1.3). ϑ Take the tensor in QK3 given by T XY = X, Yϑ(Z ) − X, Z ϑ(Y) + Z 3  X, J Yϑ(J Z ) − X, J Z ϑ(J Y) . Then T − T ϑ ∈ QK45 and we get a a a a a=1 ϑ ϑ F(T −T ) XY Z = F(T) XY Z −2T XY Z . In particular, we obtain (T −T ϑ )U2 X1 X2 = −17/10 and F(T − T ϑ )U2 X1 X2 = 18/5, so T − T ϑ ∈ QK45 \ (QK4 ∪ QK5 ). Hence the tensor S has a non-zero component in each subspace QK j, j = 1, . . . , 5. If = 1 , (λ = μ = ε = 0, δ = 1), we have S =  + T, where  ∈ QK1 , with corresponding 1-form θ = A1 + A2 , · , and T having non-zero QK3 component, with corresponding 1-form 4ϑ = A1 + A2 , · . Moreover, we get (T − T ϑ )U1 X2 X1 = −3/2 and F(T − T ϑ )U1 X2 X1 = 6 = −4(T − T ϑ )U1 X2 X1 .

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So T − T ϑ ∈ QK4 . As a computation with Maple shows, this happens for any choice of vectors. Hence S ∈ QK135 . If = 2 , (μ = δ = 0, ε = 1), then S = Sλ (λ ∈ R) decomposes as S =  + T with  ∈ QK12 \ (QK1 ∪ QK2 ). Since moreover 20ϑ = 7A1 + 2λU 1 , · , (T − T ϑ ) X1 U2 X2 = −13/20, and F(T − T ϑ ) X1 U2 X2 = 6/5, the tensor S has a non-zero component in each subspace QK j, j = 1, . . . , 5. Case 2 The space A Sp(2,1) . The forms θ i corresponding to the homogeneous quaternionic Kähler structure S = Sλ,μ (λ, μ ∈ R3 ) given in Table 2 have the following values at o ∈ A Sp(2,1) A0 θ1 2λ1 θ 2 −2λ2 θ 3 −2λ3

X1 U 1 U 1 2 0 0

0 2 0

0 0 2

X2

X2 U 2 U 2

0 0 0

0 0 0

0 0 0

0 0 0

Equation 1.4 are satisfied, with the following nonzero values of θ i at o: θ (A0 ) = 2λ1 , θ 2 (A0 ) = −2λ2 , θ 3 (A0 ) = −2λ3 , θ 1 (X1 ) = θ 2 (U 1 ) = θ 3 (U 1 ) = 2. We have S =  + T, with  ∈ QK12 \ QK1 ∪ QK2 , except for λ1 = λ2 = λ3 = 0. In this case  ∈ QK1 , with corresponding 1-form θ = 2A0 , · . As for the specific type of T inside QK345 , we first have that 10ϑ = 7A0 + λ1 X1 − λ2 U 1 − λ3 U 1 , · , so the QK3 -component of T does not vanish. To find the QK45 -component, we have for example (T − T ϑ )U2 X2 U1 = −3/10 and F(T − T ϑ )U2 X2 U1 = −3/5, so T − T ϑ is never in QK5 . Suppose now that at least one of the parameters λi , μi , i = 1, 2, 3, is non-zero. Computing now as before, we get for instance for μ2 = 0 that (T − T) A0 X2 U2 = −μ2 and F(T − T ϑ ) A0 X2 U2 = 2μ2 , which is not equal to 2(T − T ϑ ) A0 X2 U2 . This also happens for the other parameters. Hence T ∈ QK345 . If finally λi , μi , i = 1, 2, 3, are all zero, a computation with Maple shows that T ∈ QK34 (see also [6, Prop. 5.3]). To find the QK45 -component, suppose first that al least one of the parameters λi , μi , i = 1, 2, 3, is nonzero. Computing, we get for instance for γ3 = 0 that (T − T ϑ ) A1 X2 U3 = −γ3 and F(T − T ϑ ) A1 X2 U3 = 2γ3 . This also happens for the other parameters, hence the tensor S ∈ QK1345 . If all the parameters vanish, then a computation with Maple shows that S ∈ QK134 (cf. [6]). 1

Case 3 The space AG2(2) . The values of the forms θ i at o ∈ AG2(2) corresponding to the homogeneous quaternionic Kähler structure S given in Table 3 are the following

θ θ2 θ3 1

A1

A2

X1

0 0 0

0 0 0

δ/2 0 0 −3/2 0 0 0 1/2 0 0 3/2 0 0 0 1/2 0 0 −3ε/2

X2

X3

U1

U2

U3

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73

In any of the three cases, that is, = ∅, (δ = ε = 1), = 1 , (δ = 0, ε = 1), and = 2 , (δ = 1, ε = 0), we have S =  + T, with  ∈ QK12 \ (QK1 ∪ QK2 ). Since moreover one has F(T − T ϑ ) XY Z

vectors XY Z

21A1 + 11A2 , · /30 1/20

−29/160

X1 X2 X3

1

113A1 + 2A2 , · /60 49/160

−29/80

X2 X3 X1

2

72A1 + A2 , · /20

−73/320

X1 X2 X3



ϑ

∅

(T − T ϑ ) XY Z

1/20

it follows that the tensor S has a non-zero component in each primitive subspace QK j, j = 1, . . . , 5.   Acknowledgements

The authors were supported by the DGICYT, Spain, under Grant MTM

2005-00173.

References 1. Alekseevski˘ı, D.V.: Classification of quaternionic spaces with a transitive solvable group of motions. Math. USSR Izvestija 9, 297–339 (1975) 2. Alekseevsky, D.V., Cortés, V.: Homogeneous quaternionic Kähler manifolds of unimodular groups. Boll. Un. Mat. Ital. B (7) 11(2), suppl., 217–229 (1997) 3. Ambrose, W., Singer, I.M.: On homogeneous Riemannian manifolds. Duke Math. J. 25, 647– 669 (1958) 4. Bagger, J., Witten, E.: Matter couplings in N = 2 supergravity. Nuclear Phys. B 222, 1–10 (1983) 5. Castrillón López, M., Gadea, P.M., Oubiña, J.A.: Homogeneous quaternionic Kähler structures on 12-dimensional Alekseevsky spaces. J. Geom. Phys. 57, 2098–2113 (2007) 6. Castrillón López, M., Gadea, P.M., Swann, A.F.: Homogeneous quaternionic Kähler structures and quaternionic hyperbolic space. Transform. Groups 11, 575–608 (2006) 7. Cecotti, S.: Homogeneous Kähler manifolds and T-algebras in N = 2 supergravity and superstrings. Comm. Math. Phys. 124, 23–55 (1989) 8. Cortés, V.: Alekseevskian spaces. Differential Geom. Appl. 6, 129–168 (1996) 9. Fino, A.: Intrinsic torsion and weak holonomy. Math. J. Toyama Univ. 21, 1–22 (1998) 10. Fré, P., Gargiulo, F., Rosseel, J., Rulik, K., Trigiante, M., Van Proeyen, A.: Tits-Satake projections of homogeneous special geometries. Classical Quantum Gravity 24, 27–78 (2007) 11. Gadea, P.M., Oubiña, J.A.: Reductive homogeneous pseudo-Riemannian manifolds. Monatsh. Math. 124, 17–34 (1997) 12. Gordon, C.S., Wilson, E.N.: Isometry groups of Riemannian solvmanifolds. Trans. Amer. Math. Soc. 307, 245–269 (1988) 13. Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998) 14. Kiriˇcenko, V.F.: On homogeneous Riemannian spaces with an invariant structure tensor. Dokl. Akad. Nauk SSSR 252, 291–293 (1980) 15. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley, New York (1969) 16. Meessen, P.: Homogeneous Lorentzian spaces admitting a homogeneous structure of type T1 ⊕ T3 . J. Geom. Phys. 56, 754–761 (2006) 17. Montesinos Amilibia, A.: Degenerate homogeneous structures of type S1 on pseudoRiemannian manifolds. Rocky Mount. J. Math. 31, 561–579 (2001) 18. Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds. Cambridge University Press, London (1983)

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19. de Wit, B., Van Proeyen, A.: Special geometry, cubic polynomials and homogeneous quaternionic spaces. Comm. Math. Phys. 149, 307–333 (1992) 20. Witte Morris, D.: Cocompact subgroups of semisimple Lie groups. Contemp. Math. 110, 309– 313 (1990) 21. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1965)

Math Phys Anal Geom (2009) 12:75–95 DOI 10.1007/s11040-008-9052-9

The L p Spectrum of Locally Symmetric Spaces with Small Fundamental Group Andreas Weber

Received: 22 October 2007 / Accepted: 21 November 2008 / Published online: 11 December 2008 © Springer Science + Business Media B.V. 2008

Abstract We determine the L p spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L p spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L2 case, the restriction on the injectivity radius is not needed. Keywords Discrete subgroups of Lie groups · Heat semigroup on L p spaces · Laplace-Beltrami operator · Locally symmetric space · L p spectrum Mathematics Subject Classifications (2000) Primary 58J50 · 11F72 · Secondary 53C35 · 35P05

1 Introduction In this paper we study the L p spectrum σ ( M, p ), p ∈ [1, ∞), of the LaplaceBeltrami operator on a complete locally symmetric space M = \X where the universal covering X of M is a symmetric space of non-compact type with rank(X) = 1. Whether the L p spectrum of a complete Riemannian manifold M depends on p or not is related to the geometry of M. More precisely, Sturm proved

A. Weber (B) Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstr. 2, 76128 Karlsruhe, Germany e-mail: [email protected]

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in [18] that the L p spectrum is p-independent if the Ricci curvature of M is bounded from below and the volume of balls in M grows uniformly subexponentially (with respect to their radius). This is for example true if M is compact or if M is the n-dimensional euclidean space Rn . On the other hand, if the Ricci curvature of M is bounded from below and the volume density of M grows exponentially in every direction then the L p spectrum actually depends on p. More precisely, Sturm showed that in this case inf Re σ ( M,1 ) = 0 whereas inf σ ( M,2 ) > 0. An example where this happens is M = Hn , the n-dimensional hyperbolic space. In the latter case and for more general non-compact hyperbolic manifolds of the form M = \Hn where  denotes a geometrically finite discrete subgroup of the isometry group of Hn such that either M has finite volume or M is cusp free, the L p spectrum was completely determined by Davies, Simon, and Taylor in [8]. They proved that σ ( M, p ) coincides with the union of a parabolic region P˜ p and a (possibly empty) finite subset {λ0 , . . . , λm } of R0 that consists 2 of eigenvalues for  M, p . Note, that we have P˜ 2 = [ (n−1) , ∞). 4 Taylor generalized this result in [20] to symmetric spaces X of non-compact type, i.e. he proved that the L p spectrum of X coincides with a certain parabolic region P p (now defined in terms of X) that degenerates in the case p = 2 to the interval [||ρ||2 , ∞), where a definition of ρ can be found in Section 2.2. He also showed that the methods from [8] can be used in order to prove the following: Proposition 1.1 (cf. Proposition 3.3 in [20]) Let X denote a symmetric space of non-compact type and M = \X a locally symmetric space with either finite volume or injectivity radius bounded away from zero. If σ ( M,2 ) ⊂ {λ0 , . . . , λm } ∪ [||ρ||2 , ∞),

(1)

where λ j ∈ [0, ||ρ||2 ) are eigenvalues of finite multiplicity, then we have for p ∈ [1, ∞): σ ( M, p ) ⊂ {λ0 , . . . , λm } ∪ P p . However, the L2 spectrum of locally symmetric spaces (with infinite volume) is in general unknown. Moreover, for non-compact locally symmetric spaces \X with finite volume the assumption (1) is in general not fulfilled: If X is a symmetric space of non-compact type and  ⊂ Isom0 (X) an arithmetic subgroup such that the quotient M = \X is a complete, non-compact locally symmetric space, the continuous L2 -spectrum of M contains the interval [||ρ||2 , ∞) but is in general strictly larger. Nevertheless, some results concerning the L p -spectrum can be obtained if the Q-rank of  equals one (cf. [23]). Our main concern in this paper is to prove that for certain complete locally symmetric spaces M = \X (with infinite volume) the L p -spectrum of the respective Laplace-Beltrami operator coincides with the L p -spectrum of the

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77

Fig. 1 The parabolic region P p if p  = 2

universal covering X of M, i.e. with the parabolic region P p from above for any p ∈ [1, ∞): Theorem 1.2 Let X = G/K denote a symmetric space of non-compact type with rank(X) = 1 and dim(X)  3. Assume, that the injectivity radius of the complete locally symmetric space M = \X is bounded away from zero and that  ⊂ G is small. Then we have for p ∈ [1, ∞): σ ( M, p ) = P p = σ ( X, p ). The precise definitions of a small subgroup and the parabolic region P p will be given in Definition 2.18 and Definition 3.1 below. In the case p = 42 the parabolic region P p is indicated in Fig. 1 as the hatched area. Proposition 3.3 below implies that we can omit the conditions on the injectivity radius and the dimension in the case p = 2: In Corollary 3.5 we will prove that for any complete locally symmetric space M = \X, where X denotes a symmetric space of non-compact type with rank one and  ⊂ G = Isom0 (X) is a small subgroup, the following holds: σ ( M,2 ) = [||ρ||2 , ∞) = σ ( X,2 ). This generalizes a result due to K. Corlette (cf. Theorem 4.2 in [5]) who proved that the bottom of the L2 spectrum λ0 (M) coincides with ||ρ||2 , and therefore with the bottom of the L2 spectrum of the universal covering X.

2 Preliminaries 2.1 Heat Semigroup on L p Spaces In this section M denotes an arbitrary complete Riemannian manifold. The Laplace-Beltrami operator  M := −div(grad) with domain Cc∞ (M) (the set of differentiable functions with compact support) is essentially self-adjoint

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and hence, its closure (also denoted by  M ) is a self-adjoint operator on the Hilbert space L2 (M). Since  M is positive, − M generates a bounded analytic semigroup e−t M on L2 (M) which can be defined by the spectral theorem for unbounded self-adjoint operators. The semigroup e−t M is a submarkovian semigroup (i.e., e−t M is positive and a contraction on L∞ (M) for any t  0) and we therefore have the following: (1) The semigroup e−t M leaves the set L1 (M) ∩ L∞ (M) ⊂ L2 (M) invariant and hence, e−t M | L1 ∩L∞ may be extended to a positive contraction semigroup T p (t) on L p (M) for any p ∈ [1, ∞]. These semigroups are strongly continuous if p ∈ [1, ∞) and consistent in the sense that T p (t)| L p ∩Lq = Tq (t)| L p ∩Lq . (2) Furthermore, if p ∈ (1, ∞), the semigroup T p (t) is a bounded analytic semigroup with angle of analyticity θ p  π2 − arctan 2|√p−2| . p−1 For a proof of (1) we refer to [7, Theorem 1.4.1]. For (2) see [15]. In general, the semigroup T1 (t) needs not be analytic. However, if M has bounded geometry T1 (t) is analytic in some sector (cf. [6, 21]). In the following, we denote by − M, p the generator of T p (t) (note, that  M =  M,2 ) and by σ ( M, p ) the spectrum of  M, p . Furthermore, we will write e−t M, p for the semigroup T p (t). Because of (2) from above, the L p -spectrum σ ( M, p ) has to be contained in the sector   π z ∈ C \ {0} : | arg(z)|  − θ p ∪ {0} 2   | p − 2| ⊂ z ∈ C \ {0} : | arg(z)|  arctan √ ∪ {0}, 2 p−1 which is indicated in Fig. 1. If we identify as usual the dual space of L p (M), 1  p < ∞, with p L (M), 1p + p1 = 1, the dual operator of  M, p equals  M, p and therefore we always have σ ( M, p ) = σ ( M, p ). 2.2 Symmetric Spaces Let X denote a symmetric space of non-compact type with rank(X) = 1. Then G := Isom0 (X) is a semi-simple Lie group that acts transitively on X and X = G/K, where K ⊂ G is a maximal compact subgroup of G. We denote the respective Lie algebras by g and k. Given a corresponding Cartan involution θ: g → g we obtain the Cartan decomposition g = k ⊕ p of g into the eigenspaces of θ. The subspace p of g can be identified with the tangent space TeK X. We assume, that the Riemannian metric ·, · of X in p ∼ = TeK X coincides with the restriction of the Killing form B(Y, Z ) := tr(adY ◦ adZ ), Y, Z ∈ g, to p. For any maximal abelian subspace a ⊂ p we refer to = (g, a) as the set of restricted roots for the pair (g, a), i.e. contains all α ∈ a∗ \ {0} such that gα := {Y ∈ g : ad(H)(Y) = α(H)Y for all H ∈ a} = {0}.

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These subspaces gα = {0} are called root spaces. Once a positive Weyl chamber a+ in a is chosen, we denote by + the subset 1 of positive roots and by ρ := 2 α∈ + (dim gα )α half the sum of the positive roots (counted according to their multiplicity). 2.2.1 Horocyclic Coordinates The choice of a positive Weyl chamber a+ determines an Iwasawa decomposition G ∼ = N × A × K. This yields a diffeomorphism μ : N × A → X, (n, a) → na · x0 .   + We remind the reader of the fact N = exp g + β , where denotes the β∈ set of positive restricted roots for the pair (g, a) with respect to the Weyl chamber a+ . Note also, that the root spaces gα and gβ for α = β ∈ + are orthogonal with respect to the scalar product ·, · . We choose for each α ∈ + an orthonormal basis of gα and the union {N 1 , . . . , Nn−1 } of these bases yields an orthonormal basis of the subspace n := β∈ + gβ . Furthermore, we take a unit vector H from the 1-dimensional subalgebra a. This leads to a global coordinate map ϕ : X = N A · x0 → Rn , n−1 

xi Ni exp (yH) · x0 → (x1 , . . . , xn−1 , y) =: (x, y). exp i=1

We call these coordinates horocyclic coordinates since the orbits N · x are usually called horocycles in X. In the rank-1 case, the orbits under N are horospheres but for higher rank symmetric spaces this is not true as horospheres always have codimension one. 2.2.2 The Metric in Horocyclic Coordinates For all α ∈ + wedefine a left  invariant bilinear form hα on the nilpotent  subgroup N = exp g β∈ + β by  ·, · , on gα , hα := 0, else, where Y, Z := −B(Y, θ Z ) denotes the usual Ad(K)-invariant bilinear form of g. Then we have the following formula for the pullback μ∗ g of the metric g on X to N × A. Proposition 2.1 Denoting by log a the unique element in a with exp(log a) = a, the pullback μ∗ g is given by  1 −2α(log a) 2 ds(n,a) = e hα ⊕ da2 . 2 + α∈

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Proof The proof follows directly from Proposition 1.6 in [2] or Proposition 4.3 in [3] respectively.   We immediately obtain the following three important corollaries. Corollary 2.2 The representation of the metric g with respect to horocyclic coordinates is given by ⎛ −1 −2α1 (log a) ⎞ 2 e .. ⎜ 0⎟ . ⎜ ⎟ (gij)i, j(na · x0 ) = ⎜ ⎟, ⎝ ⎠ 2−1 e−2αn−1 (log a) 0

1

where the roots αi ∈ + appear in the matrix above according to their multiplicity and the indices are chosen in a way such that 

−2α(log a) e hα (Ni , Ni ) = e−2αi (log a) . α∈ +

Corollary 2.3 Let X denote a rank-1 symmetric space of non-compact type. The volume form of X with respect to horocyclic coordinates is  (n−1)/2  1 det(gij)(na · x0 ) dxdy = e−2ρ(log a) dxdy 2  (n−1)/2 1 e−2yρ(H) dxdy, = 2 where log a = yH. Corollary 2.4 Let X denote a rank-1 symmetric space of non-compact type. If we choose in the definition of horocyclic coordinates H ∈ a+ with ||H|| = 1, the Laplace-Beltrami operator in these coordinates is given by  X = −2

n−1

i=1

e2αi

∂2 ∂2 ∂ + 2||ρ|| − , ∂ y ∂ y2 ∂ xi2

where e2αi is short hand for the function (x, y) → e2yαi (H) . Proof Remember the formula X = −

1 ∂i (gij det(gij)1/2 ∂ j) 1/2 det(gij) i, j

and notice that ρ(H) = ||ρ||. Then a straightforward calculation shows the assertion.  

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2.3 Poincaré Series and its Critical Exponent Let us denote by X = G/K a symmetric space of non-compact type and by  a discrete subgroup of G that acts freely on X. The resulting (complete) locally symmetric space is denoted by M = \X. A major role in the following is played by the so-called Poincaré series

P(s; x, y) := e−sd(x,γ y) γ ∈

with s ∈ (0, ∞), x, y ∈ X, and its critical exponent δ() := inf{s ∈ (0, ∞) : P(s; x, y) < ∞}. Since all γ ∈  are isometries of X, an application of the triangle inequality implies that the definition of the critical exponent δ() does not depend on the choice of the points x and y ∈ X. We further remark, that because of P(s; γ1 x, γ2 y) = P(s; x, y) for all γ1 , γ2 ∈ , the Poincaré series P(s; ·, ·) can be considered as a function on M × M. An equivalent definition for the critical exponent δ() uses the orbit counting function N(R; x, y) := #{γ ∈  : d(x, γ y)  R}. One can prove the equality δ() = lim sup R→∞

log N(R; x, y) , R

(2)

cf. [16, 19]. The critical exponent δ() is therefore a measure for the exponential growth rate of -orbits in X. It can be proved that we always have δ() ∈ [0, 2||ρ||], [14, Section 2]. Furthermore, if  is a lattice, its critical exponent δ() attains the maximal possible value 2||ρ||, [1, Theorem 7.4]. 2.4 Geodesic Compactification and Limit Sets In this subsection we denote by X = G/K always a symmetric space of noncompact type. Our main references are [10, 11]. 2.4.1 Geodesic Compactification Definition 2.5 Two unit speed geodesics c1 and c2 of X are called asymptotic if there exists a positive constant C such that d(c1 (t), c2 (t))  C

for all t  0.

This clearly defines an equivalence relation on the unit speed geodesics of X. The set of equivalence classes is called the geodesic boundary of X and is denoted by X(∞). We further put X := X ∪ X(∞).

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If c is a unit speed geodesic in X, the respective equivalence class is denoted by c(∞). For the proof of the next proposition we refer to [11, Proposition 1.2]. Proposition 2.6 Let c0 denote a unit speed geodesic in X. Then for any point x ∈ X there exists a unique unit speed geodesic c in the equivalence class of c0 with c(0) = x. Because of this proposition, the geodesic boundary X(∞) can be identified with the unit sphere Sx X in any tangent space Tx X, in particular with the unit sphere in p ∼ = TeK X. Notation (1) By cxy we denote the unit speed geodesic connecting x and y with cxy (0) = x. (2) For v ∈ Sx X we denote by cv the unit speed geodesic with cv (0) = x and c v (0) = v. (3) For x ∈ X and ξ ∈ X(∞) we denote by cxξ the unique unit speed geodesic in the equivalence class ξ with cxξ (0) = x. (4) Let c denote a unit speed geodesic with c(0) = x0 . For any ε > 0 and r > 0 we define the truncated cone (with vertex x0 , axis c, and angle ε) by C(c, ε, r) := {x ∈ X : x0 (c (0), c x0 x (0)) < ε} \ {x ∈ X : d(x0 , x)  r}. Proposition 2.7 (cf. Proposition 2.3 in [11]) There is a unique topology τ on X = X ∪ X(∞) such that: (1) Given x ∈ X and ξ ∈ X(∞) the truncated cones C(cxξ , ε, r) with ε > 0, r > 0 form a neighborhood basis for the topology at ξ . (2) The topology of X induced from τ coincides with the original topology of X and X is a dense open subset of X. This topology τ is called cone topology. Since asymptotic geodesics are mapped onto asymptotic geodesics by isometries, we also have the following well-known result. Lemma 2.8 (cf. Remark 1.17.11 in [10]) Any isometry of X extends to a homeomorphism of X. 2.4.2 Limit Sets Definition 2.9 Let  ⊂ Isom(X) denote an arbitrary subgroup of the isometry group of X. The limit set () of  is defined as () :=  · x ∩ X(∞),

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where  · x is the closure of the orbit  · x in X. Note, that the limit set () does not depend on the choice of x ∈ X (cf. [10, p. 33]). Lemma 2.10 (1) The limit set () is a closed subset of X(∞). (2) If  ⊂ Isom(X),  = {id}, is a discrete subgroup that acts freely on X, the limit set () is always non-empty. Proof The first assertion follows directly from the definition. For a proof of the second assertion we refer to [10, p. 33].   In accordance with the theory of discrete subgroups in hyperbolic geometry we make the following definition (see e.g. [17, p. 577]). Definition 2.11 A discrete subgroup  ⊂ Isom(X) that acts freely on X is called a subgroup of the first kind if () = X(∞) and of the second kind otherwise. For a subgroup of the second kind, the ordinary set () := X(∞) \ () is always a non-empty open subset of the geodesic boundary X(∞). If rank(X) = 1, () is often called region of discontinuity. The following proposition justifies this. Proposition 2.12 (cf. Proposition 8.5 in [11]) Let X denote a symmetric space of non-compact type with rank(X) = 1 and  ⊂ Isom(X) a discrete subgroup that acts freely on X. Then the set of points in X at which the action of  is properly discontinuous is X ∪ (). Example 2.13 Let X = G/K denote a rank one symmetric space of noncompact type and G = N AK an Iwasawa decomposition of the semi-simple Lie group G = Isom0 (X). A discrete subgroup  = a , a = id of the one dimensional Lie group A is always a second kind subgroup. This can be seen as follows: If x0 ∈ X denotes the basepoint for the Iwasawa decomposition and if H = log(a) ∈ a is the unique element in the Lie algebra of A with exp(H) = a, the unparameterized geodesic A · x0 is given by the image of the curve c(t) = exp(tH) · x0 . The equivalence class of the geodesic c(t) (resp. c(−t)) is denoted by c(∞) (resp. c(−∞)). Hence, the limit set () consists only of the two points c(∞) and c(−∞) ∈ X(∞).

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Since we want to work with Dirichlet fundamental domains, we present some results here. Recall, that for a discrete subgroup  ⊂ Isom(X) that acts freely on X and x0 ∈ X the set F = F(x0 ) := {x ∈ X : d(x, x0 ) < d(γ x, x0 ) for all γ ∈  \ {id}} is called Dirichlet fundamental domain (with respect to x0 ∈ X). An easy calculation shows that F is star-shaped with respect to x0 . Lemma 2.14 (cf. Corollary 3.3.2 in [4]) Let X denote a symmetric space of noncompact type with rank(X) = 1 and  ⊂ Isom(X) a discrete subgroup that acts freely on X. If F = F(x0 ) is a Dirichlet fundamental domain, the set ·F is locally finite on X ∪ (). The following two theorems have their analogues in hyperbolic geometry (see e.g. Theorem 12.1.13 and Theorem 12.1.15 in [17]). Theorem 2.15 Let X denote a symmetric space of non-compact type with rank(X) = 1 and F = F(x0 ) a Dirichlet fundamental domain for a discrete subgroup  ⊂ Isom(X) that acts freely on X. Then    γ F ∩ () , () = γ ∈

where F denotes the closure of F in X. Proof Since the ordinary set () is -invariant, it is clear that the right hand side is contained in the left hand side. In order to prove the other inclusion, we choose ξ ∈ (). Then there exists a sequence (xi )i∈N in X converging to ξ . For any i ∈ N we can find an isometry γi ∈  such that xi ∈ γi F. Since  · F is locally finite on X ∪ (), only finitely many isometries of the sequence (γi )i are distinct. Hence, there is a j ∈ N such that xi ∈ γ j F for infinitely many i ∈ N. Therefore, we have ξ ∈ γ j F and in conclusion   ξ ∈ γ j F ∩ () = γ j F ∩ () .   Theorem 2.16 Let X denote a symmetric space of non-compact type with rank(X) = 1 and  ⊂ Isom0 (X) a subgroup of the second kind. Then vol (\X) = ∞.

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Proof Let F = F(x0 ) be a Dirichlet fundamental domain for . It clearly suffices to prove vol (F) = ∞. The ordinary set () is non-empty and homeomorphic to an open subset of the unit sphere in any tangent space. In particular, the ordinary set is a Baire space. As a discrete subgroup  is countable and therefore one of the  closed subsets γ F ∩ () of () must have non-empty interior. Thus, the interior of F ∩ () is non-empty. Choose an interior point ξ of F ∩ (). Since F is star-shaped with respect to x0 , the geodesic ray cx0 ξ : [0, ∞) → X, t → cx0 ξ (t) is completely contained in F. If U ⊂ F ∩ () is an open neighborhood of ξ , every geodesic ray starting at x0 in the equivalence class of some η ∈ U is contained in F for the same reason. Hence, there are ε > 0 and r > 0 such that the truncated cone C(cx0 ξ , ε, r) is contained in F. Let us take an Iwasawa decomposition G∼ = N× A×K of G = Isom0 (X) where K is the isotropy subgroup of G with respect to x0 , A · x0 = cx0 ξ (R), and if H ∈ a (the Lie algebra of A) is defined by the equation exp(tH) · x0 = cx0 ξ (−t), the positive Weyl chamber of a is chosen as a+ := {tH : t > 0}.

Furthermore, we choose a compact neighborhood N0 of the identity in N such that for all n0 ∈ N0 and t  r + 1 the geodesic t → n0 · cx0 ξ (t) is contained in the truncated cone C(cx0 ξ , ε, r). This is possible since μ−1 (C(cx0 ξ , ε, r) ∩ X) ⊂ N × A is an open subset of N × A that contains the set {e} × {exp(−tH) : t  r + 1} and hence, we can find a neighborhood N0 of e in N such that N0 × {exp(−tH) : t  r + 1} is contained in μ−1 (C(cx0 ξ , ε, r) ∩ X). We claim that the ‘box’ N0 · cx0 ξ (Rr+1 ) has infinite volume. Since this box is completely contained in the fundamental domain F(x0 ), this proves vol (\X) = vol (F) = ∞.

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We use horocyclic coordinates (cf. Section 2.2.1 and 2.2.2) and obtain  (n−1)/2  1 vol (N0 · cx0 ξ (Rr+1 )) = e−2yρ(H) dxdy 2 ϕ(N0 ·cx0 ξ (Rr+1 ))  (n−1)/2   −r−1 1 = dx e−2y||ρ|| dy 2 −∞ ϕ(N0 ·x0 ) = ∞.   A discrete subgroup  ⊂ Isom0 (X) that acts freely on X is called geometrically cocompact if the Kleinian manifold M := (X ∪ ()) /  is compact. In [5, Theorem 5.2] K. Corlette proved the following result. Theorem 2.17 Let X denote a symmetric space of non-compact type with rank(X) = 1 and  ⊂ Isom0 (X) a geometrically cocompact subgroup. Then, the Hausdorff dimension of the limit set () is bounded from above by the critical exponent δ(). We can conclude for geometrically cocompact subgroups  with δ()  ||ρ|| that the (closed) limit set () does not have full Hausdorff dimension. Consequently,  is a subgroup of the second kind. We do not know whether a small critical exponent δ() always implies that  is a subgroup of the second kind. Therefore, we make the following definition. Definition 2.18 Let X be a symmetric space of non-compact type with rank(X) = 1. A subgroup  ⊂ Isom0 (X) of the second kind is called small if δ()  ||ρ||. Example 2.19 (i) The subgroup  = a in Example 2.13 is small. To see this, it only remains to check whether its critical exponent is small. Recall, that the critical exponent of  can be defined as follows:  

−sd(x0 ,an x0 ) e <∞ , δ() := inf s > 0 : n∈Z

where x0 ∈ X denotes the basepoint that corresponds to the maximal compact subgroup K. If H ∈ a is the unique element in the Lie algebra a of A such that a = exp H, we obtain d(x0 , an x0 ) = d(x0 , exp(nH)x0 ) = |n| · ||H||

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and therefore for any s > 0

e−sd(x0 ,a

n

x0 )

n∈Z

=



e−s|n|·||H|| =

n∈Z

1 + e−s||H|| < ∞. 1 − e−s||H||

In particular, we may conclude δ() = 0. on the n-dimensional hyperbolic (ii) We first note that we have ||ρHn || = n−1 2 space Hn . We choose a totally geodesic submanifold Hk ⊂ Hn (with k < n) and a lattice  ⊂ Isom0 (Hk ). Then,  acts also on Hn and is obviously of the second kind (with respect to Hn ). Because of δ() = 2||ρHk || = k − 1 (see Section 2.3) the subgroup  is a small subgroup (with respect to Hn ) if and only if k  n+1 . 2 3 L p Spectrum In this section we denote by X = G/K a symmetric space of non-compact type with rank(X) = 1. We further assume as above that the restriction of the metric to the tangent space TeK X ∼ = p of the basepoint x0 = eK equals the restriction of the Killing form to p. Definition 3.1 For p ∈ [1, ∞) we put     4||ρ||2 1 y2 P p := z = x + iy ∈ C : x  1− + p p 4||ρ||2 (1 − 2p )2 if p = 2 and P2 := [||ρ||2 , ∞). Notice, that the boundary ∂ P p of the parabolic region P p is given by the curve     4||ρ||2 1 2 2 R → C, s → 1− + s + 2i||ρ||s 1 − p p p    2||ρ|| 2||ρ|| + is 2||ρ|| − − is . = p p One part of the proof of Theorem 1.2 is a consequence of Proposition 1.1. The constraint on the dimension of X and that the subgroup  is of the second kind will not be important in this part: Proposition 3.2 Let X = G/K denote a symmetric space of non-compact type with rank(X) = 1,  ⊂ G a discrete subgroup of G that acts freely on X such that the injectivity radius of M = \X is bounded away from zero and δ()  ||ρ||. Then we have for p ∈ [1, ∞): σ ( M, p ) ⊂ P p .

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Proof Corlette proved in [5] that the bottom of the L2 -spectrum of M coincides with ||ρ||2 , and hence σ ( M,2 ) ⊂ [||ρ||2 , ∞). In view of Proposition 1.1 this is enough to prove the assertion.

 

The following proposition is a crucial step towards the proof of the reverse inclusion in Theorem 1.2. Proposition 3.3 Let X = G/K denote a symmetric space of non-compact type with rank(X) = 1,  ⊂ G a subgroup of the second kind, and M = \X the respective locally symmetric space. Then, for any p ∈ [1, ∞), the boundary ∂ P p of the parabolic region P p is contained in the approximate point spectrum of  M, p : ∂ P p ⊂ σapp ( M, p ). Remark 3.4 We know that e−t M,1 : L1 (M) → L1 (M) is an analytic semigroup if the manifold M has bounded geometry (cf. [6, 21]). A consequence of Proposition 3.3 is that the semigroup e−t M,1 is not a bounded analytic semigroup—in contrast to the semigroups e−t M, p for p ∈ (1, ∞)—since the imaginary axis is tangent to the parabolic region P1 .

Since compact perturbations of any complete non-compact Riemannian manifold M do not change the continuous L2 -spectrum of  M if considered as a subset of R, the continuous spectrum depends only on the geometry at infinity of M. This follows from the so-called decomposition principle (cf. [9]): If M0 ⊂ M is a compact manifold with boundary of the same dimension as M and  the Laplacian on M \ M0 (with Dirichlet boundary conditions), then  M and  have the same continuous spectrum. That is why the supports supp( fn ) of our sequence fn in the proof of Proposition 3.3 will leave every compact subset of M at some time. Proof of Proposition 3.3 To prove this proposition, we will construct for each z ∈ ∂ P p a sequence of differentiable functions fn ∈ L p (X) with support in an open Dirichlet fundamental domain for  such that || X, p fn − zfn || L p →0 || fn || L p

if n → ∞.

(3)

Such a sequence ( fn ) descends to a sequence of differentiable functions in L p (M) and consequently, we are done as soon as we have shown (3). Our proof is subdivided into two steps. First, we construct a box completely contained in a Dirichlet fundamental domain for , that can be easily described by using horocyclic coordinates. This box will be similar to the box constructed in the proof of Theorem 2.16. Secondly, we use horocyclic coordinates to construct

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for any z ∈ ∂ P p a sequence of differentiable functions fn with support in the constructed box satisfying (3). We choose the basepoint x0 ∈ X corresponding to the maximal compact subgroup K of G. By F = F(x0 ) we denote the Dirichlet fundamental domain with respect to x0 and by F we denote the closure of F in X = X ∪ X(∞). Since  is a subgroup of the second kind, the region of discontinuity () is a non-empty open subset of X(∞) and consequently, we can find an interior point ξ in F ∩ X(∞). Let us denote by cx0 ξ the unique unit-speed geodesic in the equivalence class ξ with cx0 ξ (0) = x0 . Note, that the ray cx0 ξ (R0 ) is contained in F as the Dirichlet fundamental domain F is star-shaped with respect to its center x0 . Since the truncated cones C(cx0 ξ , ε, r) := {x ∈ X : x0 (ξ, x) < ε} \ {x ∈ X : d(x, x0 )  r} form a neighborhood basis for the cone topology at ξ ∈ X(∞), we can find ε > 0 and r > 0 such that C(cx0 ξ , ε, r) ∩ X ⊂ F. If we choose the maximal abelian subgroup A := {exp(tH) : t ∈ R} in G such that cx0 ξ (t) = exp(−tH) · x0 , and the positive Weyl chamber a+ = {tH : t > 0} of the one dimensional Lie algebra a of A, we obtain an Iwasawa decomposition G∼ = N × A × K, with a nilpotent subgroup N of G. Note, that we have ||H|| = 1. We finally choose a neighborhood N0 ⊂ N of the identity in N such that for any n0 ∈ N0 the geodesic t → n0 exp(−tH) · x0 is contained in the truncated cone C(cx0 ξ , ε, r) if t  r + 1. Hence, the box N0 {exp(tH) : t  −(r + 1)} · x0 is contained in the fundamental domain F. We proceed with the second step. Choose    2||ρ|| 2||ρ|| z = z(s) = + is 2||ρ|| − − is ∈ ∂ P p . p p With respect to horocyclic coordinates (cf. Section 2.2.1) ϕ: X = N A · x0 → Rn ,  n−1

exp xi Ni exp (yH) · x0 → (x1 , . . . , xn−1 , y) =: (x, y) i=1

we define the sequence 

fn (x, y) := b (x)cn (y)e

2||ρ|| p +is

 y

,

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with an arbitrary function b ∈ Cc∞ (ϕ(N0 · x0 )) and a (so far arbitrary) sequence cn ∈ Cc∞ ((−∞, −(r + 1)]). Since the supports of fn are clearly contained in the Dirichlet fundamental domain F for , this sequence descends to a differentiable sequence with compact supports in M. For this reason, it suffices to perform all the calculations below in the universal covering space X of M. We begin with the calculation of  X fn using the formula for the LaplaceBeltrami operator in horocyclic coordinates derived in Corollary 2.4: n−1  2

2||ρ|| 2αi (H)y ∂  X fn (x, y) = −2 e b (x) cn (y)e( p +is)y 2 ∂ xi i=1     2||ρ|| 2||ρ|| +2||ρ||b (x) cn (y) + + is cn (y) e( p +is)y p     2 2||ρ|| 2||ρ|| 2||ρ|| −b (x) cn (y)+2 +is cn (y)+ +is cn (y) e( p +is)y . p p This leads to

n−1

 2 2||ρ|| ∂  X fn (x, y) = −2 e2αi (H)y 2 b (x) cn (y)e( p +is)y ∂ xi i=1      2||ρ|| 2||ρ|| + is c n (y)−c n (y) e( p +is)y . +b (x) zcn (y)+ 2||ρ||−2 p

Therefore, we obtain

n−1

 2||ρ|| ∂2 e b (x) cn (y)e( p +is)y  X fn (x, y)−zfn (x, y) = −2 2 ∂ xi i=1    2||ρ|| 4||ρ|| −2is cn (y)−cn (y) e( p +is)y . +b (x) 2||ρ||− p 2αi (H)y

Recall the volume form (cf. Corollary 2.3)  (n−1)/2 1 e−2||ρ||y dxdy 2 of the rank-1 symmetric space X. In the subsequent lines we estimate the L p norm of  X fn − zfn :  1p p n−1  

 2α (H)y ∂ 2  e i || X fn − zfn || L p  C1 b (x) |cn (y)| p dxdy  2 ∂ x X i i=1  +C2

|b (x)|

p

|c n (y)| p dxdy

|b (x)|

p

|c n (y)| p dxdy

X

 +C3 X

 1p  1p

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and therefore

|| X fn − zfn || L p

 p  1p  n−1   2

∂     C1  ∂ x2 b (x) dx i

i=1

 |b (x)| p dx

+C2 

|b (x)| dx

+C3

p

 1p   1p 

∞ −∞ ∞ −∞

∞ −∞

e

2 pαi (H)y

|c n (y)| p dy |c n (y)| p

|cn (y)| dy p

 1p

 1p  1p

dy

for some positive constants C1 , C2 and C3 (only depending on z and p). We also have  || fn || L p =

|b (x)| p dx

 1p 

∞

−∞

|cn (y)| p dy

 1p

.

We need to construct a sequence of functions cn such that for increasing n these functions become flatter and flatter, and the supports tend to negative infinity (because of the term e2 pαi (H)y ). More precisely, we choose a function ψ ∈ Cc∞ (R), ψ = 0, with supp(ψ) ⊂ (−2, −1) and a sequence of positive real numbers rn such that rn → ∞ (for n → ∞). We define  cn (y) := ψ

y rn

 .

Since supp(cn ) is a subset of (−2rn , −rn ), we have cn ∈ Cc∞ ((−∞, −(r + 1)]) for sufficiently large n. We claim that for any choice of b ∈ Cc∞ (ϕ(N0 · x0 )) it follows || X fn − zfn || L p →0 || fn || L p

(n → ∞).

To see this, we compute 

∞ −∞

e

2 pαi (H)y

 |cn (y)| dy = rn p

−1

−2

e2 pαi (H)rn u |ψ(u)| p du

 rn max |ψ(u)| 

 p

−1 −2

e2 pαi (H)rn u du

max |ψ(u)| p −2 pαi (H)rn e , 2 pαi (H)

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and   

∞ −∞ ∞ −∞ ∞ −∞

 |cn (y)| p dy = rn

−1 −2

|c n (y)| p dy = rn1− p



|c n (y)| p dy = rn1−2 p

|ψ(u)| p du, −1

−2



|ψ (u)| p du,

−1 −2

|ψ (u)| p du.

In conclusion

e−2αi (H)rn 1 1 || X fn − zfn || L p  ci + c˜1 + c˜2 2 , p || fn || L rn rn rn i=1 n−1

with positive constants ci , c˜1 and c˜2 (depending only on p, z, and the function b ). Since the right hand side converges to 0 if n → ∞, the proof is complete.   Corollary 3.5 Let X = G/K denote a symmetric space of non-compact type with rank(X) = 1. Assume that  ⊂ G is a small subgroup. Then the following holds for the L2 -spectrum of the locally symmetric space M = \X: σ ( M,2 ) = [||ρ||2 , ∞). Proof The inclusion [||ρ||2 , ∞) ⊂ σ ( M,2 ) is a special case of the previous proposition. Since the bottom of the L2 -spectrum λ0 (M) equals ||ρ||2 (cf. [5]), the assertion follows.   Lemma 3.6 Let X = G/K denote a symmetric space of non-compact type with dim X  3 and  ⊂ G a discrete subgroup that acts freely on X. We assume furthermore, that the injectivity radius of the locally symmetric space M = \X is bounded away from zero. Then we have for all 1  p  q < ∞: e−t M, p : L p (M) → Lq (M) are bounded linear operators (t > 0). Proof Let us denote by k(t, x˜ , y˜ ) the heat kernel of M, i.e. the integral kernel of the semigroup e−t M,2 on L2 (M). Then we have an upper bound for k of the form   ˜ x˜ ) P(s; y˜ , y˜ ), k(t, x˜ , y˜ )  C(t) P(s; x,

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where C(t) is a positive function (only depending on t and s) and

P(s; x˜ , x˜ ) = e−sd(x,γ x) , γ ∈

π(x) = x˜ , denotes the Poincaré series for  (cf. [22, Corollary 3.22] or [24]). If we choose s > 2||ρ|| (and thus s > δ()), the Poincaré series is bounded since the injectivity radius of M is bounded away from zero (cf. [22, Corollary 3.4] or [24]). Therefore, for fixed t, the heat kernel k is bounded. This implies that e−t M,1 : L1 (M) → L∞ (M) are bounded linear operators. Since the same holds also for the linear operators e−t M,1 : L1 (M) → L1 (M) and T∞ (t) : L∞ (M) → L∞ (M) (cf. Section 2.1) an application of the Riesz-Thorin interpolation theorem concludes the proof.   Lemma 3.7 Let X = G/K denote a symmetric space of non-compact type with dim X  3 and  ⊂ G a discrete subgroup that acts freely on X. We assume furthermore, that the injectivity radius of the locally symmetric space M = \X is bounded away from zero. Then we have for all 1  p  q < ∞ and t > 0: e−t M, p  M, p ⊂  M,q e−t M, p . Proof We choose an f ∈ dom( M, p ) = dom(e−t M, p  M, p ). Because of e−t M, p f ∈ Lq (M) ∩ dom( M, p ) and the consistency of the semigroups e−t M, p , p ∈ [1, ∞), we have e−s M,q e−t M, p f = e−(t+s) M, p f and obtain by using Lemma 3.6: 1 || (e−s M,q e−t M, p f − e−t M, p f ) − e−t M, p  M, p f || Lq s 1  C || (e−s M, p f − f ) −  M, p f || L p → 0 (s → 0+ ). s Thus, the function e−t M, p f is contained in the domain of  M,q and we also have the equality e−t M, p  M, p f =  M,q e−t M, p f.   Proposition 3.8 Let X = G/K denote a symmetric space of non-compact type with dim X  3 and  ⊂ G a discrete subgroup that acts freely on X. We assume furthermore, that the injectivity radius of the locally symmetric space M = \X is bounded away from zero. Then we have for all 1  p  q  2: σ ( M,q ) ⊂ σ ( M, p ). Proof Using Lemma 3.7 this follows as in [13, Proposition 3.1], [12, Proposition 2.1], or [23, Proposition 3.3].  

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We are now prepared to prove Theorem 1.2: Proof of Proposition 1.2 In view of Proposition 3.2 we need only show the inclusion P p ⊂ σ ( M, p ),

p ∈ [1, ∞).

But this follows for p ∈ [1, 2] from Proposition 3.3 and Proposition 3.8 by observing  ∂ Pq . Pp = q∈[ p,2]

The remaining cases p  2 follow by duality as the parabolic regions P p and   P p coincide if 1p + p1 = 1.

References 1. Albuquerque, P.: Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28, MR MR1675889 (2000d:37021) (1999) 2. Borel, A.: Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geom. 6, 543–560, MR MR0338456 (49 #3220) (1972) 3. Borel, A.: Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7, 235–272, MR MR0387496 (52 #8338) (1974) 4. Bowditch, B.H.: Geometrical finiteness with variable negative curvature. Duke Math. J. 77(1), 229–274, MR MR1317633 (96b:53056) (1995) 5. Corlette, K.: Hausdorff dimensions of limit sets. I. Invent. Math. 102(3), 521–541, MR MR1074486 (91k:58067) (1990) 6. Davies, E.B.: Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21(2), 367–378, MR MR1023321 (90k:58214) (1989) 7. Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge, MR MR1103113 (92a:35035) (1990) 8. Davies, E.B., Simon, B., Taylor, M.E.: L p spectral theory of Kleinian groups. J. Funct. Anal. 78(1), 116–136, MR MR937635 (89m:58205) (1988) 9. Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46(3), 497–503, MR MR544241 (80j:35075) (1979) 10. Eberlein, P.B.: Geometry of nonpositively curved manifolds. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago, MR MR1441541 (98h:53002) (1996) 11. Eberlein, P.B., O’Neill, B.: Visibility manifolds. Pacific J. Math. 46, 45–109, MR MR0336648 (49 #1421) (1973) 12. Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in L p (Rν ) is p-independent. Comm. Math. Phys. 104(2), 243–250, MR MR836002 (87h:35247) (1986) 13. Hempel, R., Voigt, J.: On the L p -spectrum of Schrödinger operators. J. Math. Anal. Appl. 121(1), 138–159, MR MR869525 (88i:35114) (1987) 14. Leuzinger, E.: Critical exponents of discrete groups and L2 -spectrum. Proc. Amer. Math. Soc. 132(3), 919–927, MR MR2019974 (2004j:22011) (electronic, 2004) 15. Liskevich, V.A., Perel’muter, M.A.: Analyticity of sub-Markovian semigroups. Proc. Amer. Math. Soc. 123(4), 1097–1104, MR MR1224619 (95e:47057) (1995) 16. Nicholls, P.J.: The ergodic theory of discrete groups. In: London Mathematical Society Lecture Note Series, vol. 143. Cambridge University Press, Cambridge, MR MR1041575 (91i:58104) (1989) 17. Ratcliffe, J.G.: Foundations of hyperbolic manifolds. In: Graduate Texts in Mathematics, vol. 149. Springer, New York, MR MR1299730 (95j:57011) (1994)

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18. Sturm, K.-T.: On the L p -spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2), 442–453, MR MR1250269 (94m:58227) (1993) 19. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50, 171–202, MR MR556586 (81b:58031) (1979) 20. Taylor, M.E.: L p -estimates on functions of the Laplace operator. Duke Math. J. 58(3), 773– 793, MR MR1016445 (91d:58253) (1989) 21. Varopoulos, N.Th.: Analysis on Lie groups. J. Funct. Anal. 76(2), 346–410, MR MR924464 (89i:22018) (1988) 22. Weber, A.: Heat kernel estimates and L p -spectral theory of locally symmetric spaces. Dissertation, Universitätsverlag Karlsruhe (2006) 23. Weber, A.: L p -spectral theory of locally symmetric spaces with Q-rank one. Math. Phys. Anal. Geom. 10(2), 135–154, MR MR2342629 (2007) 24. Weber, A.: Heat kernel bounds, Poincaré series, and L2 spectrum for locally symmetric spaces. Bull. Austral. Math. Soc. 78, 73–86 (2008)

Math Phys Anal Geom (2009) 12:97–107 DOI 10.1007/s11040-008-9053-8

The Dynamics of Pendulums on Surfaces of Constant Curvature P. Coulton · R. Foote · G. Galperin

Received: 7 November 2007 / Accepted: 21 November 2008 / Published online: 14 January 2009 © Springer Science + Business Media B.V. 2009

Abstract We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases: first a pivot that moves with constant speed along a geodesic, and then a pivot that undergoes acceleration along a fixed geodesic. Keywords Dynamical systems · Geodesic · Constant curvature · Pendulum Mathematics Subject Classifications (2000) 37J · 53A · 70E · 85

1 Introduction In [1], the first and third authors investigated the motion of barbells on surfaces of constant curvature. It is natural to extend this study to pendulums. Consider a moving particle on a surface. An unconstrained particle would, of course, move along a geodesic with constant speed. The point of our work is

P. Coulton (B) · G. Galperin Mathematics Department, Eastern Illinois University, Charleston, IL, USA e-mail: [email protected] G. Galperin e-mail: [email protected] R. Foote Wabash College, Crawfordsville, IN, USA e-mail: [email protected]

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to investigate the motion of a particle constrained to be at a constant distance from a point with a given motion. A pendulum problem on a surface of constant curvature is defined as a pivot point and a mass connected to the pivot by a rigid massless rod of fixed length δ. Assume that the pivot is constrained to move along some fixed curve with prescribed motion. The rod provides the only force on the mass in order to keep the mass a fixed distance from the pivot. No torque is applied to the rod, and so the acceleration of the mass is always in the direction of the rod. Figure 1 above illustrates the motion of a pendulum. The mass m is at A and the pivot is at T. The rigid rod is the segment AT. Our results concern the case in which the pivot moves along a geodesic. We let θ denote the angle between the rod and the geodesic. We shall prove the following: Theorem A Assume that the pivot of a pendulum moves with constant speed v along a geodesic on a surface with constant curvature K. Let θ(t) be the angle that the rigid rod makes at time t with respect to the direction of pivot motion. Then the pendulum satisfies the non-linear differential equation d2 θ = −v 2 K sin(θ) cos(θ ). dt2

(1)

The generic solutions are periodic. Moreover, the period is approximately √ 2π v |K| when θ is sufficiently close to a stable equilibrium for all t. If K > 0, then there are stable equilibria at θ = 0 and θ = π and unstable equilibria at θ = π/2 and θ = 3π/2. If K < 0, then there are unstable equilibria at θ = 0 and θ = π and stable equilibria at θ = π/2 and θ = 3π/2. This differential equation indicates that the dynamics of a pendulum on a curved surface are sensitive to the absolute motion of the pivot, in contrast to the Euclidean case. It is somewhat surprising that the length of the pendulum does not play a role. Reparameterizing by the distance s = vt traveled by the pivot, the differential equation can be rewritten as d2 θ = −K sin(θ) cos(θ ). ds2

Fig. 1 The motion of a pendulum

m

A

θ T

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This shows that the dynamics of the pendulum do not depend on the magnitude of the speed of the pivot, but rather on the distance traveled by the pivot and the fact that its speed is constant. From this perspective, the period of the motion should be thought of as a distance, as opposed to a time. For smallamplitude oscillations √ sufficiently close to a stable equilibrium, the period is approximately 2π/ |K|. On a sphere, this is the circumference, and it follows that the mass of the pendulum is approximately following a geodesic slightly offset from the geodesic path of the pivot, that is, it approximately follows a path it would take were it not connected to the pivot. On a hyperbolic plane, the period is a natural distance identified by the pendulum dynamics. In contrast to the spherical case, the mass does not follow an approximate geodesic. Instead, it oscillates about a constant distance curve. Theorem B Assume that the pivot of a pendulum moves with constant speed v along a geodesic on a surface with constant curvature K. Let θ(t) be the angle that the rigid rod makes at time t with respect to the direction of pivot motion. If the pendulum√starts at a stable equilibrium with initial intrinsic angular speed dθ/dt|t=0 = v |K|, then it monotonically and asymptotically approaches an unstable equilibrium as t → ∞. More specifically, if K > 0,

θ(0) = 0,

and

√ dθ   = v K, dt t=0

then θ is increasing and lim θ(t) =

t→∞

π . 2

Similarly, if K < 0,

θ(0) =

π , 2

and

√ dθ   = v −K, dt t=0

then θ is increasing and lim θ(t) = π.

t→∞

It is interesting to note √ that when K > 0 and the motion takes place on a sphere of radius R = 1/ K in R3 , the initial intrinsic angular speed in Theorem B is equal to the constant extrinsic angular speed of the pivot: √ v K = v/R = ω. As a consequence of the Proofs of Theorems A and B, we derive a conservation law for pendulum motion. We also consider the dynamics of the pendulum when the pivot is allowed to accelerate. A barbell in a space of constant curvature K consists of a pair of masses joined by a massless rigid rod of fixed length d. We say that the barbell is balanced if the masses are equal. Consider motion of a balanced barbell such

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that its center of mass moves along a geodesic. Note that the motion of a pendulum can also be considered as the motion of half of a balanced barbell. If the entire motion of the barbell is along the geodesic, we say that it moves with a direct-motion. When the balanced barbell moves such that the rod is at a right angle to the direction of motion, we shall refer to the motion as a right-motion. If the direction of the rod from the center of mass to the leading mass makes an angle θ (θ = 0, π/2) with respect to the geodesic of motion, we shall refer to the motion of the barbell as skew-motion. In earlier work [1], the first and third authors showed that the direct-motion and right-motion of a balanced barbell are free (unconstrained) motions (Fig. 2). Theorem C If the midpoint of a balanced barbell moves along a geodesic on a surface of constant curvature with constant speed, then the motion of the individual masses is symmetric and identical to the motion of the equivalent pendulum problem. We will assume the following convention for spaces of constant curvature K: if K > 0, √  √  √  cos K (s) = cos s K , sin K (s) = sin s K , tan K (s) = tan s K and if K < 0, then

√  cos K (s) = cosh s −K , √  tan K (s) = tanh s −K .

√  sin K (s) = sinh s −K ,

Theorem D Assume that the pivot of a pendulum moves with speed v(t) along a geodesic on a surface with constant curvature K. Let θ(t) be the angle that the rigid rod makes with the direction of the motion of the pivot. Then the pendulum satisfies the differential equation  d2 θ 2 = −Kv(t) cos θ(t) sin θ(t) + |K|a(t) cot K δ sin θ(t), dt2 where a(t) = dv/dt is the scalar acceleration. This has equilibria at θ = 0, π . They are stable when K > 0 and unstable when K < 0.

geodesic geodesic

a

geodesic

b

Fig. 2 a Direct-motion, b right-motion, c skew-motion

c

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We analyze the small-amplitude oscillations in the special case when v(t) = v0 + at

2 Pendulum Motion for Constant Speed Pivots Figure 3 below illustrates additional quantities which will be used in the Proof of Theorem A. We let b denote the distance from m to the geodesic and we let a denote the distance from the pivot T to the point D. Let δ be the distance from the pivot to the mass, i.e., the length of the rod. We observe that the geodesic triangle ADT is a right triangle with right angle at D. We let α be the angle at A. The set {a1 , a2 } is an orthonormal frame at the position of the mass. Note that in the case K > 0 we must require that δ < π R to guarantee that the mass and the rod remain on the same side of the geodesic line. Proof of Theorem A Case I: K > 0√Let δ be the length of the rigid rod. We assume that δ < π R, where R = 1/ K, and take the motion to be on the sphere x2 + y2 + z2 = R2 √ in R3 . The angular speed of the  pivot is ω = v/R = v K, and the location of the pivot can be taken to be R cos(ωt)ˆi + sin(ωt)ˆj . Letting θ be the angle between the direction of motion of the pivot and the rigid rod, the possible locations of the mass are parameterized by    F(t, θ) = R sin K δ sin θ kˆ + cos θ − sin(ωt)ˆi + cos(ωt)ˆj   +R cos K δ cos(ωt)ˆi + sin(ωt)ˆj . For a particular motion of the mass we have θ = θ(t), and the motion of the mass is σ (t) = F(t, θ(t)).

Fig. 3 Skew-motion of a pendulum

a1 A

m α

a2

δ

b θ D

a

T

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The acceleration in R3 is then 2  d2 σ dθ = − R sin K δ sin(θ(t))kˆ + cos(θ (t)) 2 dt dt   × − sin(ωt)ˆi + cos(ωt)ˆj +

  dθ ω R sin K δ sin(θ(t)) cos(ωt)ˆi + sin(ωt)ˆj − dt   − ω2 R sin K δ cos(θ(t)) − sin(ωt)ˆi + cos(ωt)ˆj −

+2

  − ω2 R cos K δ cos(ωt)ˆi + sin(ωt)ˆj +

 d2 θ + R sin δ cos(θ(t))kˆ − sin(θ (t)) K dt2   × − sin(ωt)ˆi + cos(ωt)ˆj . We take {a1 , a2 } to be an oriented, orthonormal frame for the tangent space of the sphere at the point A (the location of the mass) such that a2 is perpendicular to the rigid rod at A. For a general location of the mass we have a2 = ∂∂θF / ∂∂θF , which for the mass at location σ (t) becomes   a2 (t) = cos(θ(t))kˆ − sin(θ(t)) − sin(ωt)ˆi + cos(ωt)ˆj . If g(, ) denotes the metric on the tangent space of the sphere, then  2the com ponent of d2 σ/dt2 tangent to the sphere and normal to the rod is g ddtσ2 , a2 (t) , which simplifies to  2  2

dσ dθ 2 g , a2 (t) = R sin K δ + ω sin(θ (t)) cos(θ (t)) . dt2 dt2 Since force on the mass is applied only along the rigid rod,  2 the intrinsic  d σ g dt2 , a2 (t) = 0, which yields the differential equation d2 θ Kv 2 2 = −ω sin(θ(t)) cos(θ(t)) = − sin(2θ(t)). dt2 2 Note that θ = 0, π/2, π , 3π/2 are equilibrium solutions. Making the substitution u = 2θ, the equation becomes d2 u = −Kv 2 sin u. dt2

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This is the equation of a planar pendulum in Euclidean space subject to a constant gravitational field, which is known to √ be periodic using energy considerations. The small-amplitude period is 2π/v K. It has a stable equilibrium at u = 0 mod 2π , corresponding to θ = 0, π . Similarly, it has an unstable equilibrium at u = π mod 2π , corresponding to θ = π/2, 3π/2. Case II: K < 0 For simplicity, we assume the motion takes place in the hyperbolic plane of curvature K. We use the hyperboloid model in R3 , x2 + y2 − z2 =

1 , K

z > 0,

which inherits its geometry from the Minkowski inner product g((a, b , c), (x, y, z)) = ax + b y − cz. √ √ Similar to Case I, set R = 1/ −K  and ω = v/R = v −K, and assume the pivot moves along the geodesic R sinh(ωt)ˆi + cosh(ωt)kˆ . The circle of radius δ about the origin of the model is parameterized by ˆ c(θ) = R sin K δ(cos θ ˆi + sin θ ˆj) + R cos K δ k. Translating this circle to one centered at the pivot by means of the Minkowski isometry with matrix ⎛ ⎞ cosh(ωt) 0 sinh(ωt) ⎜ ⎟ 1 0 A(t) = ⎝ 0 ⎠ sinh(ωt) 0 cosh(ωt) yields the general position of the mass: F(t, θ) = A(t)c(θ)    = R sin K δ sin θ ˆj + cos θ sinh(ωt)kˆ + cosh(ωt)ˆi   +R cos K δ cosh(ωt)kˆ + sinh(ωt)ˆi . From this point the proof is nearly identical to that of Case I with the substitution of hyperbolic functions for circular ones in the appropriate places, the permutation of the basis vectors, and the use of the Minkowski inner product. The reversal of the stable and unstable equilibria result from the change in sign of K. More specifically, consider the substitution u = 2θ + π . This changes the differential equation to d2 u = Kv 2 sin u = −|K|v 2 sin u, dt2 and the reasoning can proceed as in Case I.



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Proof of Theorem B By Theorem A, θ satisfies d2 θ Kv 2 = −Kv 2 sin θ cos θ = − sin 2θ. 2 dt 2

(2)

If K > 0, let u = 2θ. If K < 0, let u = 2θ + π . In both cases the equation becomes d2 u = −|K|v 2 sin u, dt2 which is the equation of a planar pendulum of length 1 in Euclidean space subject to a constant gravitational field of magnitude g = |K|v 2 . Its stable and unstable equilibria at u = 0 and u = π correspond to the two stable and two unstable equilibria of the pendulum on the surface, respectively. The velocity du/dt at the stable equilibrium of the system required to have the pendulum mass approach the unstable equilibrium asymptotically satisfies the energy equation

2 m du  = 2m|K|v 2 ,  2 dt t=0 i.e., the kinetic energy at t = 0 is equal to the change in potential energy as t → ∞. Solving for du/dt yields  du   = 2v |K|, t=0 dt

or

 dθ   = v |K|. t=0 dt

This completes the Proof of Theorem B.



3 Pendulum Conservation Law on Surfaces of Constant Curvature Proposition 1 Under the hypothesis of Theorem A, the following combination of intrinsic angular speed and position is constant: 

dθ 2 dt

2 − Kv 2 cos 2θ.

(3)

Proof We consider a change of variables in the expression (2) as in the Proofs of Theorems A and B. If K > 0, let u = 2θ. If K < 0, let u = 2θ + π . In both cases the equation becomes d2 u = −|K|v 2 sin u, dt2 which is the equation of a planar pendulum of length 1 in Euclidean space subject to a constant gravitational field of magnitude g = |K|v 2 . The kinetic

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energy of this pendulum is (m/2)(du/dt)2 . Taking the potential energy to be zero when u = π/2, the potential energy is −m|K|v 2 cos u. Their sum,  m du 2 − m|K|v 2 cos u, 2 dt is constant. Changing back to θ and simplifying, this becomes (3). Proof of Theorem C It should now be clear that the solutions of the constant speed pivot pendulums are symmetric with respect to the stable equilibria on a surface of constant curvature (see also [2–4]). Now we consider the case of the balanced barbell, such that the center of mass moves with constant speed along a geodesic. The symmetry of solutions for the pendulum imply that the masses can be treated independently and that if we do so, the solutions for the motion of the independent masses will lie at a fixed distance δ = d/2 from the pivot (which corresponds to the center of mass) and that a geodesic from one mass to the other will always pass through the pivot. This proves the theorem.

4 Pendulum Motion for Accelerated Pivots Proof of Theorem D Now consider the case of pendulum motion on a surface of curvature K when the pivot accelerates along a geodesic. Let s(t) denote the arc-length parameter of the pivot at time t. When K > 0, the equation of motion of the mass is given by σ (t) as above. (c.f. Proof of Theorem A). Once again, let {a1 , a2 } be an oriented orthonormal frame for the tangent space of the surface at the point A such that a2 is perpendicular to the rigid rod at the point A. We have   a2 (t) = cos(θ(t))kˆ − sin(θ(t)) − sin(s(t)/R)ˆi + cos(s(t)/R)ˆj . Again, the force   on the mass acts in the direction of the rod, and so 2 2 g d σ/dt , a2 (t) = 0, which simplifies to  R sin K δ

d2 θ v(t)2 + sin θ(t) cos θ(t) − a(t) sin θ(t) cos K δ = 0, dt2 R2

where v(t) = ds/dt and a(t) = d2 s/dt2 are the speed and scalar acceleration of the pivot. This can be rewritten as √ d2 θ 2 = −Kv(t) cos θ(t) sin θ(t) + Ka(t) cot K δ sin θ(t), dt2 where cot K δ = cos K δ/ sin K δ. When K < 0 the computation is similar, and one gets √ d2 θ = −Kv(t)2 cos θ(t) sin θ(t) + −Ka(t) cot K δ sin θ(t), 2 dt

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and so the general formula is  d2 θ = −Kv(t)2 cos θ(t) sin θ(t) + |K|a(t) cot K δ sin θ(t). 2 dt The equilibria at θ = 0, π are still present, and by linearizing one easily sees that they are still stable when K > 0 and unstable when K < 0. The other equilibria, however, are gone because the other angles for which d2 θ/dt2 = 0 now depend on t. This proves Theorem D. The formula above should be compared with the formula in Theorem A. Note that if a(t) = g and K → 0, the equation reduces to the standard pendulum equation for constant acceleration, g d2 ψ + sin(ψ) = 0, 2 dt δ where ψ = θ − π . For additional insight, assume that s(t) = v0 t + at2 /2, where v0 and a are constant. We have    d2 θ 2 = −K(v + at) cos θ(t) + |K|a cot δ sin θ(t). 0 K dt2 When K > 0, linearizing near θ = 0 yields   √ d2 θ ≈ −K(v0 + at)2 + Ka cot K δ θ(t). 2 dt Linearizing near θ = π and replacing θ − π by ψ yields   √ d2 ψ 2 ≈ −K(v + at) − Ka cot δ ψ(t). 0 K dt2 These are similar to a pendulum in Euclidean space subject to an increasingly strong gravitational force. One surmises that the resulting oscillations have decreasing amplitude and period, something borne out by computer experiments. When K < 0, we analyze the situation more heuristically. We have   √ d2 θ 2 = −K(v + at) cos θ(t) + −Ka cot δ sin θ(t). 0 K dt2 √ Let θ0 (t) ∈ [0, π ] satisfy −K(v0 + at)2 cos θ0 (t) + −Ka cot K δ = 0, which is possible for all sufficiently large t. We see that θ0 (t) is smooth and approaches π/2 monotonically from above as t → ∞. Assume that θ(t) is close to θ0 (t) and set ψ(t) = θ(t) − θ0 (t). Assuming t is large, we make the approximations sin θ0 (t) ≈ 1 and θ0 (t) ≈ 0. With these approximations, linearizing in ψ yields d2 ψ ≈ K(v0 + at)2 ψ. dt2 This is similar to a pendulum in Euclidean space subject to an increasingly strong gravitational force. One surmises that the resulting oscillations have decreasing amplitude and period, and that the center of the oscillations tend toward π/2 from above, which is also observed in computer experiments. 

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References 1. Coulton, P., Galperin, G.: Forces along equidistant particle paths. Math. Phys. Anal. Geom. 7, 187–192 (2004) 2. Nagy, P.: Dynamical invariants of rigid body motion on the hyperbolic plane. Geom. Dedicata 37, 125–139 (1991) 3. Salvai, M.: On the dynamics of a rigid body in the hyperbolic space. J. Geom. Phys. 36, 126-139 (2000) 4. Zitterbarth, J.: Some remarks on the motion of a rigid body in a space of constant curvature without external forces. Demonstratio Math. XXIV(3–4), 465–494 (1991)

Math Phys Anal Geom (2009) 12:109–115 DOI 10.1007/s11040-008-9054-7

Ultraweak Continuity of σ -derivations on von Neumann Algebras Madjid Mirzavaziri · Mohammad Sal Moslehian

Received: 26 March 2008 / Accepted: 29 December 2008 / Published online: 30 January 2009 © Springer Science + Business Media B.V. 2009

Abstract Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ -derivation on a von Neumann algebra M. We show that there are a surjective ultraweakly continuous ∗-homomorphism  : M → M and a derivation D : M → M such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ -derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ -derivation on M, then there is an ultraweakly continuous linear mapping  : M → M such that d is a ∗--derivation. Keywords Derivation · ∗-homomorphism · ∗-σ -derivation · Inner σ -derivation · von Neumann algebra · Ultraweak topology · Weak (operator) topology Mathematics Subject Classifications (2000) Primary 46L57 · Secondary 46L05 · 47B47

M. Mirzavaziri · M. S. Moslehian (B) Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran e-mail: [email protected] M. Mirzavaziri e-mail: [email protected] M. Mirzavaziri · M. S. Moslehian Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran

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1 Introduction In 1932 von Neumann and Murray put quantum theory on a firm theoretical basis, i.e. operator algebras. Their work was motivated by the theory of unitary group representations and certain aspects of the quantum mechanical formalism. The von Neumann algebras, as the most important operator algebras, are applied in mathematical physics since the most fruitful algebraic reformulation of quantum statistical mechanics and quantum field theory was in terms of these algebras. The study of theory of derivations in operator algebras is motivated by questions in quantum physics and statistical mechanics, cf. [1, 2]. There are some applications of σ -derivations to develop an approach to deformations of Lie algebras which have many applications in models of quantum phenomena and in analysis of complex systems; cf. [4]. Let A and B be two algebras, X be a B-bimodule and σ : A → B be a linear mapping. A linear mapping d : A → X is called a σ -derivation if d(ab ) = d(a)σ (b ) + σ (a)d(b ) for all a, b ∈ A. These maps have been extensively investigated in pure algebra. Recently, they have been treated in the Banach algebra theory; see [3, 6, 11] and references therein. A wide range of examples are as follows: (i) Every ordinary derivation of an algebra A into an A-bimodule X is an ιA -derivation (throughout the paper, ιA denotes the identity map on the algebra A); (ii) Every endomorphism α on A is a α2 -derivation; (iii) For a given homomorphism ρ on A and a fixed arbitrary element X in an A-bimodule X, the linear mapping δ(A) = Xρ(A) − ρ(A)X is a ρ-derivation of A into X which is said to be an inner ρ-derivation. (iv) Every point derivation d : A → C at the character θ is a (θ, θ )-derivation. In [7] the present authors investigated the continuity of σ -derivations. As a consequence, they showed that every σ -derivation on a von Neumann algebra is continuous, if σ is continuous. So it is reasonable to assume that all of our mappings are norm continuous. See also [5] for an approach to the continuity of generalized derivations without linearity as well as [8]. In this paper, we investigate the automatic ultraweak continuity of σ derivations in relation to the ultraweak continuity of the mapping σ and show that if σ is an ultraweakly continuous surjective ∗-linear mapping, then every σ derivation is automatically ultraweakly continuous. The importance of our approach is that σ is a linear mapping in general, not necessarily a homomorphism. Throughout the paper, M is a von Neumann algebra acting on a Hilbert space H with orthonormal basis {eλ }λ∈I . By the weak (operator) topology on B(H) we mean the topology generated by the semi-norms T → | Tξ, η | (ξ, η ∈ H). We also use the terminology ultraweak (operator) topology for the weak∗ -topology on B(H) considered as the dual space of the nuclear operators on H. When we speak of the ultraweak continuity (the weak continuity, resp.) of a mapping on M, we mean that we equipped M with the

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ultraweak topology (weak topology, resp.). We refer the reader to [1, 2, 10] for undefined phrases and notations.

2 Main Results In this section, we investigate automatic the ultraweak continuity for σ -derivations with respect to the ultraweak continuity of the mapping σ . Theorem 2.1 Let σ be a linear mapping and d be an ultraweakly continuous ∗-σ -derivation on a von Neumann algebra M. Then there is an ultraweakly continuous linear mapping  : M → M such that d is a ∗--derivation. Proof For fixed A ∈ M define ϕ A : B (H) → B (H) by ϕ A (B) = d(A)B. Then ϕ A is an ultraweakly continuous linear mapping on B (H). To see this let {Bγ }γ ∈ be a net in B (H) with ultraweak − limγ Bγ = 0. We have     lim ϕ A (Bγ )h, h = lim d(A)Bγ h, h γ

γ

  = lim Bγ h, d(A)∗ h γ

= 0. Moreover, ker ϕ A is a right ideal of B (H). Let B ∈ ker ϕ A and C ∈ B (H). Then ϕ A (BC) = d(A)BC = ϕ A (B)C = 0. Thus I = A∈M ker ϕ A is an ultraweakly closed right ideal of B (H). We can therefore deduce that there is a projection Q ∈ I such that I = QB (H) (see [10, Proposition II.3.12]). Since Q ∈ I, Q ∈ ker ϕ A for each A ∈ M. Thus d(A)Q = ϕ A (Q) = 0 for all A ∈ M. This implies that d(A∗ )Q = 0 or equivalently Qd(A) = 0. Hence Pd(A) = Pd(A) + Qd(A) = d(A) = d(A)P + d(A)Q = d(A)P = d(A), where P = I − Q. This shows that P is the range projection onto the closed linear span of A∈M d(A)H and so P ∈ M. Define  : M → M by (A) = σ (A)P. Then d is a -derivation. We have d(AB) = d(AB)P = d(A)σ (B)P + σ (A)d(B)P = d(A)(B) + σ (A)Pd(B) 

= d(A)(B) + (A)d(B).

We claim that QH = A∈M ker d(A).  Let Qh ∈ QH. Thus d(A)Qh = = 0 for all A ∈ M . Hence Q H ⊆ ϕ A (Q)h A∈M ker d(A). On the other hand,  if h ∈ A∈M ker d(A) then d(A)h = 0 for all A ∈ M. Let B be the operator defined on H by  h if λ = λ0 Beλ = 0 otherwise

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where λ0 is a fixed element of I. Then ϕ A (B) = d(A)B = 0 for all A ∈ M. So B ∈ QB (H) and hence B = QC  for some C ∈ B (H). Thus⊥h = Beλ0 = QCeλ0 ∈ shows that Q H = QH. This A∈M ker d(A) and so QH is the closed linear  span of A,B∈M ϕ A (B)(H), i.e. the closed linear span of d(M)(H). We finally show that  is ultraweakly continuous on M. First note that d is weakly continuous on any bounded subset of M, since the weak topology and the ultraweak topology on bounded subsets of M coincide. Next let {Cγ }γ ∈ be a net in the closed unit ball of M with ultraweak − limγ Cγ = 0. Hence weak − limγ Cγ = 0. Fix h, h ∈ H. Since Ph ∈ P(H) = Q(H)⊥ , for given ε > 0 there is N d(An )hn  h   ε for some A1 , · · · , A N an N ∈ N such that σ  Ph − n=1 in M and h1 , · · · , h N in H. Hence     | (Cγ )h, h | = | σ (Cγ )Ph, h | 

 N        σ (Cγ ) Ph − d(An )hn , h    n=1  N

       σ (Cγ )d(An )hn , h  +   n=1   N       σ   Ph − d(An )hn  h    n=1  N

       + (d(Cγ An ) − d(Cγ )σ (An ))hn , h    n=1  N

        ε+ (d(Cγ An ) − d(Cγ )σ (An ))hn , h  .   n=1

  Taking limit on γ we have (Cγ )h, h → 0, since d is weakly continuous on bounded subsets. Therefore  is weakly continuous and hence ultraweakly continuous on the closed unit ball of M and thus it is ultraweakly continuous on M, by a well-known results deduced from the Krein–Smulian Theorem.

We can also prove similar facts as in Section 3 of [7] in the case of ultraweak continuity as follows. Theorem 2.2 Let σ be a ∗-linear mapping and let d be an ultraweakly continuous σ -derivation on a von Neumann algebra M. Then there is an ultraweakly continuous linear mapping  : M → M such that d is a -derivation. Proof Let ϕ A , I, Q and P be as in Theorem 2.1 and ψ A : B (H) → B (H) be defined by ψ A (B) = Bd(A). Then ψ A is an ultraweakly continuous  linear mapping on B (H) whose kernel is a left ideal of B (H). Thus J = A∈M ψ A

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is a ultraweakly closed left ideal of B (H). So there is a projection Q ∈ J such that J = B (H)Q . Hence Q d(A) = ψ A (Q ) = 0 for all A ∈ M. We thus have P d(A) = P d(A) + Q d(A) = d(A) = d(A)P + d(A)Q = d(A)P = d(A), where P = I −Q . This shows that P is the range projection onto the closed linear span of A∈M d(A)H and so P ∈ M. Define  : M → M by (A) = P σ (A)P. Then d is a -derivation. We have d(AB) = P d(AB)P = P d(A)σ (B)P + P σ (A)d(B)P = d(A)P σ (B)P + P σ (A)Pd(B) = d(A)(B) + (A)d(B). By the same way as in the above theorem we can prove that  is ultraweakly continuous on M.

By the same argument as in Theorem 2.1 one can prove the following. Proposition 2.3 Let σ be an ultraweakly continuous ∗-linear mapping and d be a σ -derivation on a von Neumann algebra M. Then there are an ultraweakly continuous ∗-homomorphism  : M → M and a -derivation D : M → M such that D is ultraweakly continuous if and only if so is d. Moreover, if d preserves ∗, then so does D. Remark 2.4 In the notation of Proposition 2.3, if σ is surjective, then  : M → M Q is also surjective. By the same strategy as in the proof of Theorem 2.2, one can prove the following result. Proposition 2.5 Let σ be an ultraweakly continuous linear mapping and d be a ∗-σ -derivation on a von Neumann algebra M. Then there are an ultraweakly continuous ∗-homomorphism  : M → M and a ∗--derivation D : M → M such that D is ultraweakly continuous if and only if so is d. The following lemma is due to S. Sakai (personal communication). Lemma 2.6 Let ρ be an ultraweakly continuous ∗-epimorphism and δ be a ∗ρ-derivation on a von Neumann algebra M. Then there are a central projection P ∈ M and an ∗-isomorphism ρ˜ : M P → M such that δ : M P → M is a ∗-ρ-derivation. ˜ Moreover, δ(A) = 0 for each A ∈ M(I − P). Also, ρ˜ −1 ◦ δ is an ordinary derivation on M P.

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Proof Since ρ is an ultraweakly continuous ∗-homomorphism, its kernel is an ultraweakly closed ideal of M. Hence there is a central projection Q ∈ M such that ker ρ = M Q. Set P = I − Q. Then ρ˜ = ρ|M P is a ∗-isomorphism. We have δ(ABP) = δ(APBP) = δ(AP)ρ(AP) + ρ(AP)δ(BP). Hence δ is a ∗-ρ-derivation ˜ on M P. Moreover, if A = B(I − P) ∈ M(I − P) is a projection then   δ(A) = δ (BQ)2 = δ(BQ)ρ(BQ) + ρ(BQ)δ(BQ) = 0, since BQ ∈ M Q = ker ρ. The space M(I − P) is a von Neumann algebra, because it is ker ρ = ρ −1 ({0}) and ρ is ultraweakly continuous. Thus M(I − P) is generated by its projection and so δ(A) = 0 for each A ∈ M(I − P). The last assertion is now obvious.

Theorem 2.7 Let σ be an ultraweakly continuous surjective ∗-linear mapping on a von Neumann algebra M. Then every σ -derivation d : M → M is automatically ultraweakly continuous. Proof By Proposition 2.3 and Remark 2.4, we may assume that σ is an ultraweakly continuous ∗-epimorphism. By Lemma 2.6, σ˜ −1 ◦ d is an ordinary derivation. By [9, Theorem 2.2.2] we conclude that σ˜ −1 ◦ d is ultraweakly continuous. Hence so is d = σ˜ ◦ (σ˜ −1 ◦ d).

Proposition 2.5 implies the following assertion. Theorem 2.8 Let σ be an ultraweakly continuous surjective linear mapping on a von Neumann algebra M. Then every ∗-σ -derivation on M is automatically ultraweakly continuous.

References 1. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 1. C∗ - and W ∗ -algebras, symmetry groups, decomposition of states, 2nd edn. Texts and Monographs in Physics. Springer, New York (1987) 2. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics, 2nd edn. Texts and Monographs in Physics. Springer, Berlin (1997) 3. Brešar, M., Villena, A.R.: The noncommutative Singer-Wermer conjecture and φ-derivations. J. Lond. Math. Soc. (2) 66(3), 710–720 (2002) 4. Hartwig, J., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using σ -derivations. J. Algebra 295(2), 314–361 (2006) 5. Hejazian, S., Janfada, A.R., Mirzavaziri, M., Moslehian, M.S.: Achievement of continuity of (ϕ, ψ)-derivations without linearity. Bull. Belg. Math. Soc.-Simon Stevn. 14(4), 641–652 (2007) 6. Mirzavaziri, M., Moslehian, M.S.: σ -derivations in Banach algebras. Bull. Iran. Math. Soc. 32(1), 65–78 (2006) 7. Mirzavaziri, M., Moslehian, M.S.: Automatic continuity of σ -derivations in C∗ -algebras. Proc. Am. Math. Soc. 134(11), 3319–3327 (2006)

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8. Mirzavaziri, M., Moslehian, M.S.: A Kadison-Sakai type theorem. Bull. Aust. Math. Soc. (in press) 9. Sinclair, A.M., Smith, R.R.: Hochschild Cohomology of von Neumann Algebras. Cambridge University Press, Cambridge (1995) 10. Takesaki, M.: Theory of Operator Algebras. I, Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences. 124. Operator Algebras and Non-commutative Geometry, 5. Springer, Berlin (2002) 11. Zhan, J.M., Tan, Z.S.: T-local derivations of von Neumann algebras. Northeast. Math. J. 20(2), 145–152 (2004)

Math Phys Anal Geom (2009) 12:117–139 DOI 10.1007/s11040-008-9055-6

Multi-particle Anderson Localisation: Induction on the Number of Particles Victor Chulaevsky · Yuri Suhov

Received: 27 November 2008 / Accepted: 29 December 2008 / Published online: 30 January 2009 © Springer Science + Business Media B.V. 2009

Abstract This paper is a follow-up of our recent papers Chulaevsky and Suhov (Commun Math Phys 283:479–489, 2008) and Chulaevsky and Suhov (Commun Math Phys in press, 2009) covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum N-particle system on a lattice Zd with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. Fröhlich and Spencer, Commun Math Phys 88:151–184, 1983; Fröhlich et al., Commun Math Phys 101:21–46, 1985; von Dreifus and Klein, Commun Math Phys 124:285–299, 1989) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript (Chulaevsky and Suhov 2007). Recently, Aizenman and Warzel (2008) proved spectral and dynamical localisation for N-particle lattice systems with a short-range interaction, using an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in Aizenman and Molchanov (Commun Math Phys 157:245–278, 1993) and Aizenman et al. (Commun Math Phys 224:219–253, 2001) (see also

V. Chulaevsky (B) Département de Mathématiques et d’Informatique, Université de Reims, Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France e-mail: [email protected] Y. Suhov Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

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references therein) which is also combined with an induction on the number of particles. Keywords Anderson localisation · Multi-particle systems Mathematics Subject Classifications (2000) Primary 47B80 · 47A75 · Secondary 35P10

1 Introduction and the Main Result The status of the multi-particle Anderson localisation problem has been described in [3], Section 1.1; the reader is advised to consult this reference. The configuration space of the N-particle lattice system is the Cartesian product Zd × · · · × Zd of N copies of a cubic lattice Zd , which we denote for brevity by Z Nd . The Hilbert space of the N-particle lattice system is 2 (Z Nd ). (N) The Hamiltonian H = HU,V,g (ω) is a lattice Schrödinger operator acting on functions φ ∈ 2 (Z Nd ) by H(N) φ(x) = H 0 φ(x) + (U(x) + gW(x; ω))φ(x)    φ(y) + U(x) + gW(x; ω) φ(x), = y∈Z Nd : y−x=1

where

W(x; ω) =

N j=1

(1.1) V(x j; ω),

x = (x1 , . . . , x N ), y = (y1 , . . . , y N ) ∈ Z Nd . Here and below we use boldface letters such as x, y, H, etc., referring to a multi-particle system, where the particle number enters as an index or specified verbally. For example, small-case boldface letters x, y, etc., will stand for designate points in Z Nd , called N-particle configurations. Letters x, y will be systematically used for points in Zd or Rd , referred to as single-particle positions (or briefly, positions). Our proof of N-particle Anderson localisation is organised as an induction in N, as has been explained in earlier presentations (see, e.g., [5]). Thus, we will have to deal with systems with smaller number of particles, 1  n < N. The respective objects, viz., points in Znd , n < N, are still denoted by boldface letters: x ∈ Znd , y ∈ Znd , etc.   (d)  (d)  Next, x j = x(1) and y j = y(1) stand for the positions of j , . . . , xj j , . . . , yj d individual particles in Z , j = 1, . . . , N, and  ·  denotes the sup-norm: for v = (v1 , . . . , v N ) ∈ Rd × · · · × Rd = R Nd , v = max v j, j=1,2

(1.2)

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where, for v = (v(1) , . . . , v(d) ) ∈ Rd ,

v = max v(i) . i=1,...,d

(1.3)

We will consider the distance on R Nd , Z Nd and Rd , Zd generated by the norm  · . Throughout this paper, the random external potential V(x; ω), x ∈ Zd , is assumed to be real IID, with a common CDF FV on R. The condition on FV guaranteeing the validity of our results is as follows:

   1 sup sup FV (a + ) − FV (a) < +∞, (1.4) A a∈R ∈(0,1)  for some A > 0. In other words, the marginal distribution of the random potential is Hölder-continuous.1 Clearly, this does not require the absolute continuity of FV . Parameter g ∈ R is traditionally called the coupling, or amplitude, constant. The interaction energy function U is assumed to be of the form    Φ x j1 , x j2 , x = (x1 , . . . , x N ) ∈ Z Nd , (1.5) U(x) = 1 j1 < j2  N

where function Φ : Zd × Zd → R (the two-body interaction potential) satisfies the following properties. (i) Φ is a bounded symmetric function:     sup |Φ x, x | : x, x ∈ Zd < +∞,

    Φ x, x = Φ x , x , x, x ∈ Zd . (1.6)

(ii) Φ has a finite range:

  Φ x, x = 0,

if x − x  > r0 ,

(1.7)

where r0 ∈ [0, +∞) is a given value. It is then obvious that function U : Z Nd → R is symmetric under any permutation of positions x j: U(x) = U(Sσ x). Here σ is an arbitrary element of the symmetric group S N , and, given x = (x1 , . . . , x N ) ∈ Z Nd ,   Sσ x = xσ (1) , . . . , xσ (N) . The same is true for function W (see Eq. 1.1). We consider binary interaction potentials in order not to make our notations excessively cumbersome. The reader will see that, actually, more general bounded short-range many-body interactions can be treated in the same way. The symmetry does not play an important role, but is convenient technically (and natural from the physical point of view).

1 One

can easily show that the main result of this paper remains valid for log-Hölder continuous CDF Fv , satisfying |FV (a + ) − FV (a)|  C ln−A ||−1 with A > 0 large enough.

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Throughout the paper, P stands for the joint probability distribution of RVs {V(x; ω), x ∈ Zd }. The main assertion of this paper is Theorem 1 Consider the random Hamiltonian H(N) (ω) given by Eq. 1.1. Suppose that U satisfies conditions (1.4) and (1.5), and the random potential {V(x; ω), x ∈ Zd } is IID obeying Eq. 1.3. Then there exists g∗ ∈ (0, +∞) such that for any g with |g|  g∗ , the spectrum of operator H(N) (ω) is P-a.s. pure point. Furthermore, there exists a nonrandom constant m+ = m+ (g) > 0 such that all eigenfunctions Ψ j(x; ω) of H(N) (ω) admit an exponential bound: |Ψ j(x; ω)|  Cj(ω) e−m+ x .

(1.8)

The assertion of Theorem 1 can also be stated in the form where ∀ given m+ > 0, ∃ g∗ = g∗ (m+ ) ∈ (0, +∞) such that ∀ g with |g|  g∗ , the eigenfunctions Ψ j(x; ω) of H(N) (ω) admit exponential bound (1.6). Remarks 1. The threshold value g∗ in Theorem 1 depends on N: g∗ = g∗ (N). (It also depends on FV and Φ.) The important question is how g∗ grows with N. We plan to address this problem in a separate paper. 2. It suffices to prove Theorem 1 for any bounded interval I ⊂ R of length  δ0 with a given, suitably chosen δ0 > 0. This is convenient (albeit not crucial) in some arguments used below. The conditions of Theorem 1 are assumed throughout the paper. As was said earlier, the Proof of Theorem 1 uses mainly MSA, in its N-particle version. The MSA scheme for N particles does not differ in principle from that for two particles; for that reason, we will often refer to paper [7]. Most of the time we work with finite-volume approximation operators (N) = H (N)(N) (ω) given by H (N) ΛL (u) ΛL (u) H(N)(N) = H(N) Λ(N) (u) + Dirichlet boundary conditions on ∂Λ(N) L (u) (1.9) ΛL (u) L (N) and acting on vectors φ ∈ CΛL (u) by

H(N)(N) φ(x) = ΛL (u)

(N) y∈ L (u): y−x=1

Λ

  φ(y) + U(x) + gW(x; ω) φ(x),

(1.10)

with the external N-particle random potential W(x; ω) as in Eq. 1.1. Here and below, Λ(N) L (u) stands for an ‘N-particle lattice box’ (a box, for short) of size L (d) d around u = (u1 , . . . , u N ), where u j = (u(1) j , . . . , uj ) ∈ Z :   N Λ(N) L (u) = × Λ L u j j=1

(1.11)

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  ∈ Zd : where Λ L (u j) is a ‘single-particle box’ around u j = u1)j , . . . , ud) j     d (i) Λ L u j = × u(i) ∩ Zd . (1.12) j − L/2, u j + L/2 i=1

For a box

Λ(N) L (u)

as in Eq. 1.11, we will also use the notation:  

jΛ(N) L (u) = Λ L u j

and (N) N

Λ(N) L (u) = ∪ j=1 j Λ L (u);

(1.13)

(N) d set ΠΛ(N) L (u) ⊂ Z describes the single-particle ‘base’ of Λ L (u). Next, ∂Λ(N) L (u) in Eq. 1.7 stands for the interior boundary (or briefly, the (N) boundary) of box Λ(N) (u): ∂Λ(N) L (u) is formed by points y ∈ Λ L (u) such that  Nd  L(N) \ Λ L (u) with y − v = 1. These definitions remain valid if ∃ a site v ∈ Z we replace N with n = 1, . . . , N − 1. As follows from Eqs. 1.7, 1.8, H (N)(N) is a Hermitian operator in the Hilbert ΛL (u) (u)). In fact, the approximation (1.7) can be used for any finite space 2 (Λ(N) L subset Λ(N) ⊂ Z Nd of cardinality |Λ(N) | and with boundary ∂Λ(N) , producing Hermitian operator H(N)(N) in 2 (Λ(N) ). Λ Hamiltonian H(N) and its approximants H(N)(N) admit the permutation symΛ metry. Namely, let Sσ be the unitary operator in 2 (Z Nd ) induced by map Sσ :

Sσ φ(x) = φ(Sσ x).

(1.14)

(N) (N) and S−1 This implies, in (N) . σ H (N) Sσ = H Λ Sσ Λ (N) particular, that for any finite Λ ⊂ Z Nd , the eigenvalues of operators H(N)(N) and H(N) (N) are identical. Λ Sσ Λ Like its two-particle counterpart (see [6, 7]), the N-particle MSA scheme involves a number of technical parameters borrowed from the single-particle MSA; see [8]. Following [8] and [6, 7], given a number α ∈ (1, 2) and starting with L0  2 and m0 > 0, we define an increasing positive sequence Lk : (N) Sσ = H(N) Then S−1 σ H

Lk = Lα0 , k

k  1,

(1.15)

and a decreasing positive sequence mk (depending on a positive number γ): mk = m0

k    −1/2 1 − γLk ,

k  1.

(1.16)

j=1

In fact, it suffices to set α = 3/2, albeit we will use the symbolic form of parameter α instead of its value: this makes our notations less cumbersome. Besides, it will make our notation agreed with that of [8]. We will also make use of parameters p = p(N, g) > d and q = q(N, p(N, g)) > p,

(1.17)

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varying with the number of particles N. The roles of parameters p and q (and the choice of their values) have been discussed in [7]: they appear systematically in the exponents of power-law bounds for probabilities of “unwanted”, or “unlikely” events defined in terms of finite-volume Hamiltonians H(N) . Λ These bounds also depend on d, α and γ (which could be added to the list of arguments for p and q) and are specified, for a given value of N, recursively, depending on the values { p(n) and q(n, p(n)) for n-particle systems, where n = 1, . . . , N − 1}. In the course of presentation, it will be made clear (and used in various places) that, for any N  1, p(n, g), q(n, g) → +∞ as |g| → ∞,

n = 1, . . . , N.

(1.18)

Note that sequence mk in Eq. 1.12 is indeed positive, and the limit lim mk  k→∞

m0 /2 when L0 is sufficiently large. (A similar observation was, in fact, made in the Appendix in [8].) We will also assume that L0 > r0 . The single-particle MSA scheme was used in [8] to check, for IID potentials, decay properties of the Green’s functions (GFs) for single-particle Hamiltonians with IID external potentials. As was said before, for a twoparticle model, the MSA scheme was established in [6, 7]. In this paper we adopt a similar strategy for the N-particle model. Here, the GFs in a box Λ(N) L (u) are defined by:    G(N)(N) (E; x, y) = ΛL (u)

H(N)(N) − E ΛL (u)

−1

δx , δy ,

x, y ∈ Λ(N) L (u),

(1.19)

where δx (v) is the lattice delta-function and ·, · stands for the scalar product in 2 (Λ(N) L (u)). Definition 1 Fix E ∈ R and m > 0. An N-particle box Λ(N) L (u) is said to be (N) (E, m)-non-singular (in short: (E, m)-NS) if the GFs G (N) (E; u, u ) defined ΛL (u) (N) by Eq. 1.15 for the Hamiltonian H (N) from Eq. 1.8 satisfy ΛL (u) (N) max G (N) (E; u, y)  e−mL . (1.20) (N) Λ (u) L y∈∂ Λ L (u) Otherwise, it is called (E, m)-singular (or (E, m)-S). A similar concept can be introduced for any finite set Λ(N) ⊂ Z Nd . Definition 2 Let n be a positive integer and J be a non-empty subset of (n) {1, . . . , n}. We say that box Λ(n) L (y) is J -separable from a box Λ L (x) (or, d equivalently, a point y ∈ Z is called (J , L)-separable from a point x) if ⎛ ⎞ ⎛ ⎞   ⎝ ⎠ ⎝ i Λ(n) (y) ∪ Λ(n) (x)⎠ = ∅.

jΛ(n) (1.21) L (y) ∩ L L j∈J

i∈J

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(n) A pair of boxes Λ(n) L (x), Λ L (y) is said to be separable (or, equivalently, a pair nd of points x, y ∈ Z is called L-separable) if, for some J ⊆ {1, . . . , n}, either (n) (n) Λ(n) L (y) is J -separable from a box Λ L (x), or Λ L (x) is J -separable from a box (n) Λ L (y).

The notion of separability of boxes is designed so as to enable us to establish Wegner–Stollmann type bounds2 (cf. [5, 12, 13]); see Eqs. 2.2, 2.3. In Lemma 1 we give a geometrical upper bound for the set of points y which are not separable from a given point x. Lemma 1 Given an n  2, let x ∈ Znd be an n-particle configuration. For any L > 1, there exists a finite collection of n-particle boxes Λ  x(l) ), l = L(l) ( (l)  1, . . . , K(x, n), K(x, n)  K(n) < ∞, of sides  L  5nL such that if a configuration y ∈ Znd satisfies y ∈

K(x,n) 

(l) Λ

(1.22)

=1 (n) then the boxes Λ(n) L (x) and Λ L (y) are separable.

The Proof of Lemma 1 is given in Appendix A. The following Theorem 2 is completely analogous to Theorem 2.3 in [8] and to Theorem 2 in [6], and so is its proof, which we omit. The reader can check, by inspecting the proofs in the single-particle case [8] and in the twoparticle case [6] that the only modification which causes concern is the choice of intermediate constants, depending on N. However, the core argument of the proof remains unchanged. Theorem 2 Let I ⊆ R be a bounded interval. Assume that for some m0 > 0 and L0 /2 > 1, lim mk  m0 /2, and for any k  0 the following properties hold: k→∞

(N) If two boxes Λ(N) Lk (u), Λ Lk (v) are separable, then   (DS.k, I, N) −2 p(N) (N) P ∀ E ∈ I : Λ(N) . Lk (u) or Λ Lk (v) is (mk , E)−NS  1 − Lk

(1.23) Here Lk and mk are defined in Eqs. 1.11, 1.12, with p, α and γ satisfying Eq. 1.14. Then, for |g| large enough, with probability one, the spectrum of operator H(N) (ω) in I is pure point. Furthermore, there exists a constant m+ 

2 For random potentials admitting a (bounded) marginal probability density, W. Kirsch has proved

an analog of Eq. 2.2, as well as the existence of the DoS for multi-particle systems.

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m0 /2 such that all eigenfunctions j(x; ω) of H(N) (ω) with eigenvalues E j(ω) ∈ I decay exponentially fast at infinity, with the effective mass m+ : | j(x; ω)|  C j(ω) e−m+ x .

(1.24)

In future, the eigenvectors of finite-volume Hamiltonians appearing in arguments and calculations, will be assumed normalised. We stress that it is the property (DS.k, I, N) encapsulating decay of the GFs which enables the N-particle MSA scheme to work. (Here and below, DS stands for ‘double singularity’). Clearly, Theorem 1.1 would be proved, once the validity of property (DS.k, I, N) is established for all k  0. Our strategy, as indicated in the title of this paper and mentioned earlier in this section, is an induction on the number of particles N  1. The base of this induction had been established earlier, starting from papers [8–10], with the help of the MSA, and also in [1, 2], in a different way, with the help of the FMM. This allows us to use results of the single-particle localisation theory. We show in this paper that, assuming a certain number of facts established for systems with n = 1, . . . , N − 1 particles, one can establish similar facts for N-particle systems. Once these facts, mostly concerning the decay properties of Green’s functions in finite boxes, are established for N-particle systems, they imply, in a fairly standard way (essentially, in the same way as in the single-particle and in the two-particle [7] theories) the spectral localization for N-particle systems. So, according to this plan, we assume established all necessary properties of n-particle systems, 1  n  N − 1, and use them whenever necessary. Of course, these properties have to be re-established for n = N. When appropriate, we discuss technical details of proofs in previous works, where the required properties have been proved for n = 1. In other words, our paper is organised as a proof of the induction step from N − 1 to N particles. Within this induction step, we use another inductive scheme—the MSA—where some properties of Green’s functions are proved first at an initial scale L0 , and then recursively derived for N-particle boxes of sizes Lk , k  1. The main property that we have to verify for a given N and for all Lk , k  0, is (DS.k, I, N). Further, the main technical parameter is the exponent p = p(N) = p(N, g) figuring in the RHS of (DS.k, I, N). At the initial step of induction in N, we use an important fact from the single-particle theory [8]: one can guarantee any (arbitrarily large) value p(1, g), provided that |g| is large enough. Cf. Eq. 1.14. Then we show that a similar property holds for any N and for k = 0, i.e., for the scale L0 (cf. Theorem 3). Therefore, in our double induction scheme (on N and, for a given N, on k), we require |g| to be sufficiently large so as to guarantee: (i) property (DS.k, I, n) for all k  0 and for n = 1, . . . , N − 1 (this property is defined verbatim, following Eq. 1.20 mutatis mutandis); (ii) property (DS.k, I, N) for k = 0.

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Parameter q = q(N) = q(N, g) is controlled via Wegner–Stollmann type bounds (WS1.n), (WS2.n) in Eqs. 2.2, 2.3, which are proved for all scales Lk at once, without induction in k.

2 The N-Particle MSA Scheme In view of Theorem 2, our aim is to check property (DS.k, I, N) in Eq. 1.20. We now outline the N-particle MSA which is used for this purpose. In both single- and N-particle versions, the MSA scheme is an elaborate scale induction in k dealing with GFs GΛ(N) (u) = G(N)(N) and involving several ΛLk (u) Lk mutually related parameters; some of them have been used in Sections 1 and 2. For a detailed discussion of the role of each parameter, see [7]. We will focus in the rest of the paper on the aforementioned scale induction in k, along sequences {(Lk , mk )} outlined in Eqs. 1.11, 1.12. Consequently, in some definitions below we refer to the particle number parameter n  1, whereas in other definitions - where we want to stress the passage from N − 1 to N - we will use the capital letter. Definition 3 Given n  1, E ∈ R, v ∈ Znd and L  2, we call the n-particle box (n) Λ(n) L (v) E-resonant (briefly: E-R) if the spectrum of the Hamiltonian H (n) ΛL (v) satisfies

  β (n) dist E, spec H (n) < e−L , where β = 1/2. (2.1) ΛL (v) Box Λ(n) L (v) is called E-completely non-resonant (briefly: E-CNR) if it is E-NR and does not contain any E-R box of size  L1/α . Throughout this paper, we use parameter β instead of its value, 1/2. As with α = 3/2, this may be helpful to readers familiar with [8] and make our notations less cumbersome. Given n  1 and L0  2, introduce the following properties (WS1.n) and (WS2.n) of random Hamiltonians H (n)(n) , l  L0 . Λl   (WS1.n) ∀ l  L0 , box Λl(n) (x) and E ∈ R: P Λl(n) (x) is not E-CNR
(WS2.n)

(n) ∀ l  L0 and separable boxes Λ(n)  (x) and Λ (y),   (2.3) P ∃ E ∈ R : neither Λl(n) (x) nor Λl(n) (y) is E−CNR
Here q = q(n) is the parameter mentioned in Eqs. 1.13, 1.14. As we already said, the initial step of the N-particle MSA scheme consists in establishing properties (S.0, I, N) and (DS.0, I, N); see Eqs. 2.4 and 1.20. The inductive step of the N-particle MSA consists in deducing

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property (DS.k + 1, I, N) from property (DS.k, I, N); again see Eq. 1.20. Both the initial and the inductive step will be done with the assistance of properties (WS1.n) and/or (WS2.n), n = 1, . . . , N, which have to be proved independently of the scale induction. In our context, properties (WS1.n) and (WS2.n) have been established in [6], Theorems 1, 2. (Despite the fact that properties (WS1.n) and (WS2.n) had been stated [6] for n = 2, their proof is automatically extended to the case of a general n.) A bound similar to (WS1.n) was independently proved by Kirsch [11]. The assumption of independence of potential V can be relaxed; see [4]. For reader’s convenience we repeat the corresponding assertion from [6]: Lemma 2 Under the above assumptions on {V(x; ω)} and U (see Eqs. 1.3–1.5), properties (WS1.n), (WS2.n) hold true ∀ positive integer n. Let I ⊆ R be an interval. Given m0 > 0 and L0  2, consider property (S.0, I, N) : (S.0, I, N)

∀ x ∈ Z Nd ,

  −2 p P ∃ E ∈ I : Λ(N) . (2.4) L0 (x) is (E, m0 )−S < L0

Here p = p(N) is the parameter mentioned in Eqs. 1.13, 1.14. The initial MSA step is summarised in the Theorem 3 below. It is completely analogous to Proposition A.1.2 in [8], and so is its proof. Note as well that multi-particle analogs of Propositions A.1.1 and A.1.3 from [8] can also be proved in the same way as in [8]. The reason for that is that the multi-particle structure of the external potential W(x; ω) and the presence of a bounded interaction potential U(x) (as well as the form of U(x) in Eq. 1.4) are virtually irrelevant for these statements. Theorem 3 ∀ given m0 and L0  2 and ∀ bounded interval I ⊂ R, there exists g0∗ = g0∗ (N, m0 , L0 , I) ∈ (0, +∞) such that for |g|  g0∗ : (A) Properties (S.0, I, N) and (DS.0, I, N) hold true. (B) Moreover, there exists a function  g:  p ∈ (d, +∞) →  g( p) ∈ [g0∗ , +∞) such that if |g|   g( p), then Eq. 2.4 is satisfied with p =  p. Equivalently, there exists a function p(N, g) of parameter g ∈ [g0∗ , +∞) (referred to in Eqs. 1.13, 1.14) such that p(N, g) → ∞ as |g| → ∞ and Eq. 2.4 is satisfied with p = p(N, g). To complete the inductive MSA step, we will prove Theorem 4 ∀ given m0 > 0, there exist g1∗ ∈ (0, +∞) and L∗1 ∈ (0, +∞) such that the following statement holds. Suppose that |g|  g1∗ and L0  L∗1 . Then, ∀ k = 0, 1, . . . and ∀ interval I ⊆ R, property (DS.k, I, N) implies (DS.k + 1, I, N).

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The Proof of Theorem 4 occupies the rest of the paper. Before we proceed further, let us repeat that the property (DS.k, I, N) for ∀ k  0 and ∀ unit interval I ⊂ R, follows directly from Theorems 3 and 4. To deduce property (DS.k + 1, I, N) from (DS.k, I, N), we introduce Definition 4 Given R > 0, consider the following set in Z Nd :   Nd D R = x = (x1 , . . . , x N ) ∈ Z : max x j1 − x j2   N R 1 j1 , j2  N

(2.5)

It is plain that, with R = r0 + 2L, if u is not in D R and x is in  L (u), then there is a subset J of {1, . . . , N} with 1  card J < N and min

j1 ∈J , j2 ∈J

x j1 − x j2  > r0 .

(N) An N-particle box Λ(N) L (u) is called fully interactive when Λ L (u) ∩ Dr0  = ∅, (N) and partially interactive if Λ L (u) ∩ Dr0 = ∅. For brevity, we use the terms an FI-box and a PI-box, respectively.

The procedure of deducing property (DS.k + 1, I, N) from (DS.k, I, N) is done here separately for the following three cases. (N) (I) Both Λ(N) Lk+1 (x) and Λ Lk+1 (y) are PI-boxes.

(N) (II) Both Λ(N) Lk+1 (x) and Λ Lk+1 (y) are FI-boxes. (III) One of the boxes is FI, while the other is PI.

These three cases are treated in Sections 3, 4 and 5, respectively. The end of Section 5 will mark the end of the Proof of Theorem 4. We repeat that all cases require the use of property (WS1.N) and/or (WS2.N).

3 Case I: Partially Interactive Pairs of Singular Boxes In this section, we aim to derive property (DS.k + 1, I, N) for a pair of (N) partially interactive and separable boxes Λ(N) Lk+1 (x), Λ Lk+1 (y). Recall, we are allowed to assume property (DS.k, I, N) for every pair of separable boxes Λ(N) x), Λ(N) y), where x, y, x, y ∈ Z Nd . In fact, we will be able to establish Lk ( Lk ( property (DS.k + 1, I, N) for partially interactive separable boxes Λ(N) Lk+1 (x),

Λ(N) Lk+1 (y) directly, without referring to (DS.k, I, N). (However, in cases (II) and (III) such a reference will be needed.)   Let Λ(N) Lk+1 (u) be an PI-box and write u = (u1 , . . . , u N ) as a pair (u , u ) where J is a non-empty subset of {1, . . . , N} figuring in Definition 4, and c u = uJ ∈ ZJ and u = uJ c ∈ ZJ are the corresponding sub-configurations in u: u = (u j, j ∈ J ) and u = (u j, j ∈ J ). Set: n = card J and n = N − n . It is convenient to represent Λ(N) L (u) as the Cartesian product (n )    (n )    Λ(N) Lk+1 (u) = Λ Lk+1 u × Λ Lk+1 u

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and write x = (x , x ) in the same fashion as (u , u ). Correspondingly, the can be written in the form Hamiltonian H(N)(N) ΛLk+1 (u)           Hφ(x) = φ(y) + U x + gW x ; ω + U x + gW x ; ω φ (x) , Λ(N) Lk+1 (u):

y∈

y−x=1

(3.1) or, algebraically, 



H (N)(N) = H (n ) (N) ⊗ I + I ⊗ H (n ) (N) . 1;Λ L (u ) 2;Λ L (u ) ΛLk+1 (u) k+1 k+1

(3.2)

Here I is the identity operator on the complementary variable. Due to the symmetry of terms U and W, in the forthcoming argument we can assume, without loss of generality, that J = {1, . . . , n },

J c = {n + 1, . . . , N}.

Definition 5 Let be n ∈ {1, . . . , N − 1}, k  0 and u = (u1 , . . . , un ) ∈ Znd .  Given a bounded interval I ⊂ R and m > 0, the n-particle box Λ(n) Lk (u ) is called m-tunneling (m-T, for short) if ∃ E ∈ I and disjoint n-particle (n) (n)  boxes Λ(n) Lk−1 (v1 ), Λ Lk−1 (v2 ) ⊂ Λ Lk (u ) which are (E, m)-S. An N-particle box 



(n ) (n )       of the form Λ(N) Lk (u) = Λ Lk−1 (u ) × Λ Lk−1 (u ), with n + n = N, u = (u , u ),     u = (u1 , . . . , un ), u = (un +1 , . . . , u N ), is called (m, n , n )-partially tunelling  (n )   ( (m, n , n )-PT) if either Λ(n Lk−1 (u ) or Λ Lk−1 (u ) is m-T. Otherwise, it is called

  (m, n , n )-NPT. Finally, a box Λ(N) Lk (u) is called m-PT if it is (m, n , n )-PT for some n , n  1 with n + n = N, and m-NPT, otherwise.

The following statement will be sometimes referred to as the NITRoNS property of PI-boxes: Non-Interacting boxes are Tunneling, Resonant or (otherwise) Non-Singular. Cf. [7]. 



(n )  (n )  Lemma 3 Consider an N-particle box Λ(N) Lk (u) of the form Λ Lk (u ) × Λ Lk (u ),      n d  n where u = (u , u ), u = (u1 , . . . , un ) ∈ Z , u = (un +1 , . . . , u N ) ∈ Z d . As  sume that ∀ j1 , j2 with 1  j1  n , n + 1  j2  N, we have u j1 − u j2  > r0 , (N) so that Λ(N) Lk (u) is PI. Assume also that Λ Lk (u) is E-CNR and m-NPT. Let        (n )   )  (λa , ϕa ) , a = 1, . . . , |Λ(n (μb , ψb ) , b = 1, . . . , |Λ Lk u | , Lk u | , 



be the eigenvalues and eigenvectors of H(n (n)  ) and H(n (n) ) , respectively. Set ΛLk (u ) ΛLk (u )   −(1−β) N(d−1) − L−1 . m = m 1 − L k k ln Lk Then we have max (n ) 1a|Λ L (u )| k



max |G(n (n )   v ∈∂ Λ L (u ) k

)



  u , v ; E − λa |  e−m Lk

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129

and, similarly, max (n ) 1b |Λ L (u )| k

    max |G(n ) u , v ; E − μb |  e−m Lk . (n ) v ∈∂ Λ L (u ) k

 Moreover, this implies that the N-particle box Λ(N) Lk (u) is (E, m )-NS.

The Proof of Lemma 3 is given in Appendix B. (It is fairly straightforward and based on the representations (7.1)–(7.3).)

Lemma 4 Let n, k be positive integers and suppose that (DS.k, I, n) holds true. Then  1  1 − 2 p(n) +2d 2 −2 p(n) P Λ(n) (y) is m-PT  |Λ(n) = Lk α . (3.3) Lk Lk (y) | Lk−1 2 2 Here p(n) is the parameter figuring in Eqs. 1.13, 1.14. Proof Combine (DS.k, I, n) with a straightforward (albeit not sharp) up2 per bound 12 |Λ(n) Lk (y) | for the number of pairs of centers v1 , v2 of boxes

(n) (n) Λ(n) Lk−1 (v1 ), Λ Lk−1 (v2 ) ⊂ Λ Lk (y).

 

In Lemma 5 we assume for simplicity that a PI box Λ(N) Lk (y) corresponds to an N-particle system that splits into two subsystems, with particles 1, . . . , n and n + 1, . . . , n + n = N, respectively, and the two subsystems do not interact with each other. Lemma 5 Let Λ(N) Lk (y) be an N-particle PI box, with (n )    (n )    Λ(N) Lk (y) = Λ Lk y × Λ Lk y ,

where n , n  1, n +n = N;  (yn +1 , . . . y N ) ∈ Zn d , and min

1in

y= (y , y ), min

n +1 j N



y= (y1 , . . . yn ) ∈ Zn d ,

y =

yi − y j > r0 .

Then for any given value p(N) > 0 there exists g2∗ ∈ (0, +∞) such that if |g|  g2∗ , then   1 −2 p(N) L P Λ(N) . (3.4) Lk (y) is m-PT  4 k Proof By Definition 5, box Λ(N) Lk (y) is m-PT iff at least one of constituent 



)  (n )  boxes Λ(n Lk (y ), Λ Lk (y ) is m-T. By Lemma 4, inequality (3.3) holds for both   n = n and n = n . Since ∀ n < N p(n, g) → ∞, this leads to the assertion of Lemma 5.  

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Remark The assertion of Lemma 5 remains true for a general type of interaction (with appropriate modifications), but is simpler and more transparent in the case of two-body interaction of the form (1.4). This explains our choice of the interaction energy function U(x). Besides, in applications to the electron transport problems, such a choice is perfectly justified: here, a commonly accepted form of interaction is two-body Coulomb. We repeat that, according to the structure of the MSA scheme, for any given number of particles n = 1, . . . , N, any (i.e., arbitrarily large) values p(n), q(n) can be used, provided that |g| is sufficiently large. In other words, parameters p(n), q(n) follow Eq. 1.14. Indeed, for p(n) this can be guaranteed, by direct inspection, for the boxes of initial size L0 . Cf. Appendix in [8]. The same property is then reproduced inductively at any scale Lk , k  1. As to q(n), one can actually obtain a stronger bound:   β −Lk P Λ(N)  L−s Lk (u) is E-R  e k for any a priori given s ∈ (0, ∞) including s = q(N), provided that β > 0 and L0 (hence, any Lk ) is large enough. Lemma 6 Assume that property (WS2.N) and Eqs. 3.3, 3.4 hold true. Suppose also that |g| is sufficiently large, so that for all n = 1, . . . , N − 1 the bound (3.3) holds with p(n)  2 p(N) + 2d, and that L0 is sufficiently large, so that for any k  0 we have 1 −2 p(N) . L 4 k Then, ∀ interval I ⊆ R, ∀ integer k  0 and ∀ pair of separable PI N-particle (N) boxes Λ(N) Lk (x) and Λ Lk (y),  1  −2 p(N) −q(N) (N) P ∃ E ∈ I : Λ(N) (x) and Λ (y) are (E, m )−S  Lk + Lk . k Lk Lk 2 (3.5) Here p(N), q(N) are the parameters from Eq. 1.13. − 2 p(n) α +2d

Lk



Proof of Lemma 6 By virtue of Lemma 3 (NITRoNS property),   (N) P ∃ E ∈ I : Λ(N) Lk (x) and Λ Lk (y) are (E, mk )−S   (N)  P Λ(N) Lk (x) or Λ Lk (y) is mk −PT +   (N) + P ∃ E ∈ I : neither Λ(N) (x) nor Λ (y) is E−CNR . Lk Lk

(3.6)

Now the assertion of Lemma 6 follows from Eqs. 3.6, 2.3 and Lemma 5.

  −2 p(N)

Remark It is readily seen that the RHS of Eq. 3.5 is bounded by Lk −q(N) −2 p(N) provided that Lk < Lk /2, i.e., for q(N) large enough.

,

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131

An immediate corollary of Lemma 6 is the following Theorem 5 ∀ given interval I ⊆ R and k = 0, 1, . . ., property (DS.k, I, N) (N) holds for all pairs of separable PI-boxes Λ(N) Lk (x), Λ Lk (y). Summarising the above argument: as was said earlier, verifying property (DS.k + 1, I, N) for a pair of N-particle PI-boxes did not force us to assume (DS.k, I, N). However, in the course of deriving (DS.k + 1, I, N) for PIboxes we used property (WS2.N). This completes the analysis of the case (I) where both boxes Λ(N) Lk+1 (x) and

Λ(N) Lk+1 (y) are PI. For future use, we also give

(N) Lemma 7 Consider a N-particle box Λ(N) Lk+1 (u). Let M = M(Λ Lk+1 (u); E) be

(l) the maximal number of (E, mk )-S, pair-wise separable PI-boxes Λ(N) Lk (u ) ⊂ Λ Lk+1 (u). The following property holds



P ∃E ∈ I :

M(Λ(N) Lk+1 (u);



E)  2 

α L2d k

 ·

 1 −2 p (N−1) −q(N) L + Lk , (3.7) 2 k

where p (N − 1, g) := min{ p(n, g),

1  n  N − 1}

−→ |g|→∞

+ ∞.

(3.8)

As before, p(N), q(N) are the parameters from in Eqs. 1.13, 1.14. Proof of Lemma 6 The number of possible pairs of centres (u(l1 ) , u(l2 ) ), 1  l1 < l2  M, is bounded by L2d k+1 /2, while for a given pair of centres one can apply Lemma 6. This leads to the assertion of Lemma 7.  

4 Fully Interactive Pairs of Singular Boxes The main outcome in case (II) is Theorem 6 placed at the end of this section. Before we proceed further, let us state a geometric assertion (see Lemma 8 below) which we prove in Section 6. Lemma 8 Let be n  1, L > r0 and consider twoseparable n-particle FI-boxes (n)  (n)  (n)   Λ(n) L (u ) and Λ L (u ), with dist Λ L (u ), Λ L (u ) > 8L. Then   (n)    ΠΛ(n) = ∅. L u ∩ ΠΛ L u

(4.1)

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Lemma 8 is used in the Proof of Lemma 9 which, in turn, is important in establishing Theorem 6. In fact, Lemma 8 is a natural development of Lemma 2.2 in [6]. Let I ⊆ R be an interval. Consider the following assertion (N) ∀ pair of interactive separable boxes Λ(N) Lk (x) and Λ Lk (y):  (IS.k.N) :  −2 p(N) (N) P ∃ E ∈ I : both Λ(N) , Lk (x), Λ Lk (y) are (E, mk )-S  Lk

(4.2) with p(N) as in Eqs. 1.13, 1.14. (This is a particular case of (DS.k, I, N)). Lemma 9 Given k  0, assume that property (IS.k.N) holds true. Consider a  (N) box Λ(N) Lk+1 (u) and let N(Λ Lk+1 (u); E) be the maximal number of (E, mk )-S, pair-

(N) ( j) wise separable FI-boxes Λ(N) Lk (u ) ⊂ Λ Lk+1 (u). Then ∀   1,    (N) (u); E)  2  L2(1+dα) · L−2p(N) . P ∃ E ∈ I : N(Λ Lk+1 k k

(4.3)

(N) (2n) (1) ) Proof of Lemma 9 Suppose ∃ FI-boxes Λ(N) Lk (u ), . . ., Λ Lk (u (N) ⊂ Λ Lk+1 (u) such that any two of them are separable. By virtue of Lemma 8, it is readily seen that (2i−1) (2i) ), Λ(N) ), the respective (random) operators (a) ∀ pair Λ(N) Lk (u Lk (u (N) (N) H (N) (2i−1) (ω) and H (N) (2i) (ω) are mutually independent, and so are ΛLk (u ) ΛLk (u ) their spectra and Green’s functions G(N)(N) (2i−1) and G(N)(N) (2i) . ΛLk (u ) ΛLk (u ) (b) Moreover, the following pairs of operators form an independent family:   (N) (N) H (N) (2i−1) (ω), H (N) (2i) (ω) , i = 1, . . . , , (4.4) ΛLk (u ) ΛLk (u )

Indeed, operator H(N)(N) (i) , with i ∈ {1, . . . , 2n}, is measurable relative to the ΛLk (u ) (N) (i) (i) sigma-algebra B (Λ(N) Lk (u ) generated by {V(x), x ∈ Π Λ Lk (u )}, i = 1, . . . , 2. (i) Now, by Lemma 4.2, the sets Π Λ(N) Lk (u ), i ∈ {1, . . . , 2}, are pairwise disjoint, (N) (i) so that all sigma-algebras B (Λ Lk (u ), i ∈ {1, . . . , 2}, are independent. Thus, any collection of events A1 , . . ., A related to the corresponding pairs   (N) (N) H (N) (2i−1) , H (N) (2i) , i = 1, . . . , , ΛLk (u ) ΛLk (u ) also form an independent family. Now, for i = 1, . . . ,  − 1, set    (2i+1)   (2i+2)  Ai = ∃ E ∈ I : Λ(N) and Λ(N) are (E, mk ) -S . Lk u Lk u

(4.5)

Then, by virtue of (IS.k.N)(see Eq. 4.3),   −2 p(N) P A j  Lk ,

(4.6)

0  j   − 1,

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133

and by virtue of independence of events A0 , . . ., An−1 , we obtain ⎧ ⎫ −1 −1 ⎨" ⎬  & '  −2 p(N)  P Aj = P A j  Lk . ⎩ ⎭ j=0

(4.7)

j=0

To complete the proof, note that the total number of different families of 2 (N) boxes Λ(N) Lk ⊂ Λ Lk+1 (u) with required properties is bounded from above by 2 2 1  1  2 (Lk /2 + r0 + 1) Ldk+1  2Lk Ldk+1  L2(1+dα) , k (2)! (2)! since their centres must belong to the subset D Lk +r0 ∩ Λ(N) Lk+1 (u) (see Eq. 2.5). Recall also that r0 < L0  Lk ∀ k  0, by our assumption and by construction. This yields Lemma 9.   Lemma 10 Let K(u, Lk+1 ; E) be the maximal number of (E, mk )-S, pair(N) ( j) wise separable boxes Λ(N) Lk (u ) ⊂ Λ Lk+1 (u) (fully or partially interactive). Then ∀   1, −2 p(N−1)

P { ∃E ∈ I : K(u, Lk+1 ; E)  2+2 }  L4dα k · Lk

−2p(N)

+ L2(1+dα) · Lk k

,

(4.8) where p(N − 1) and p(N) are parameters from Eqs. 1.13, 1.14, for the system with N − 1 and N particles, respectively. Proof of Lemma 10 Assume that K(u, Lk+1 ; E)  2 + 2. Let M(Λ(N) Lk+1 (u) ; E)

be as in Lemma 7 and N(Λ(N) Lk+1 (u) ; E) as in Lemma 9. Obviously,     (N) K (u, Lk+1 ; E)  M Λ(N) Lk+1 (u) ; E + N Λ Lk+1 (u) ; E .

(N) Then either M(Λ(N) Lk+1 (u) ; E)  2 or N(Λ Lk+1 (u) ; E)  2. Therefore,

P { ∃E ∈ I : K(u, Lk+1 ; E)  2 + 2 }    P ∃E ∈ I : M(Λ(N) ; E)  2 + Lk+1 (u)   + P ∃E ∈ I : N(Λ(N) Lk+1 (u) ; E)  2 −2 p(N−1)

 L4dα k · Lk

−2p(N)

+ L2(1+dα) · Lk k

,  

by virtue of Eqs. 3.8 and 4.3. An elementary calculation now gives rise to the following

Corollary 1 Under assumptions of Lemma 10, with   4, p(N − 1) and p(N) large enough and for L0 large enough, we have, ∀ integer k  0, −2 p(N)−1

P { ∃E ∈ I : K(u, Lk+1 ; E)  2 + 2 }  Lk+1

.

(4.9)

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Now the Wegner–Stollmann bound (WS2.N) implies (N)   Lemma 11 If N-particle boxes Λ(N) Lk+1 (u ), Λ Lk+1 (u ) (fully or partially interactive) are separable, then ∀ L0 > (J + 1)2 ,     (N)    P ∀ E ∈ I : either Λ(N) u or Λ u is (E, J)−CNR Lk+1 Lk+1 −(q(N)α −1 −2α)

 1 − (J + 1)2 Lk+1

−(q (N)−4) > 1 − Lk+1 .

(4.10)



Here q(N) is the parameter from Eq. 1.13 and q (N) := q(N)/α. The statement of Lemma 12 below is a simple reformulation of Lemma 4.2 from [8], adapted to our notations. Indeed, the reader familiar with the proof given in [8] can see that the structure of the external potential is irrelevant to this completely deterministic statement. So it applies directly to our model with potential energy U(x) + gW(x; ω). For that reason, the Proof of Lemma 12 is omitted. Lemma 12 Fix an odd positive integer J and suppose that the following properties are fulfilled:   (N) (i) Λ(N) (v) is (E, J)−CNR, and (ii) K Λ ; E  J. Lk+1 Lk+1 (u) Then for sufficiently large L0 , box Λ(N) Lk+1 (v) is (E, mk+1 )-NS with ( ) 5J + 6 mk+1  mk 1 − > m0 /2 > 0. 1/2 Lk

(4.11)

Taking into account Corollary 1, we set J = 2 + 1. Now the main result of this section: Theorem 6 Fix a bounded interval I ⊂ R. For p(N) large enough there exists L∗0 ∈ (0, +∞) such that if L0  L∗0 and p(N − 1) is large enough, then, ∀ k  0, property (IS.k.N) in Eq. 4.2 implies (IS.k + 1.N) , with the same p(N). (N) Proof of Theorem 6 Let x, y ∈ Z Nd and assume that Λ(N) Lk+1 (x) and Λ Lk+1 (y) are separable FI-boxes. Consider the following two events:   (N) B = ∃ E ∈ I : both Λ(N) Lk+1 (x) and Λ Lk+1 (y) are (E, mk+1 )-S ,

and, for a given odd integer J,   (N) R = ∃ E ∈ I : neither Λ(N) Lk+1 (x) nor Λ Lk+1 (y) is (E, J)−CNR . By virtue of Lemma 11, for L0  (J + 1)2 and α = 3/2, we have: −(q (N)−4)

P { R } < Lk+1

,

q (N) := q(N)/α.

(4.12)

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135 −q (N)+4

Further, P { B }  P { R } + P { B ∩ Rc }, and we know that P { R }  Lk+1 . So, it suffices now to estimate P { B ∩ Rc }. Within the event B ∩ Rc , for any E ∈ (N) I, either Λ(N) Lk+1 (x) or Λ Lk+1 (y) must be (E, J)-CNR. Without loss of generality, assume that for some E ∈ I, Λ(N) Lk+1 (x) is (E, J)-CNR and (E, mk+1 )-S. By

Lemma 12, for such value of E, K(Λ(N) Lk+1 (x); E)  J + 1. We see that   B ∩ Rc ⊂ ∃E ∈ I : K(Λ(N) Lk+1 (x); E)  J + 1 and, therefore, by Lemma 10, with J = 2 + 2, and Corollary 1,

  & ' −2 p(N) P B ∩ Rc  P ∃E ∈ I : K(Λ(N) (x); E)  J + 1  Lk . Lk+1

(4.13)  

Remark The integer J figuring throughout Section 4 depends on N, d, and the choice of parameter p(N). In turn, p(N) is determined by dimension d and the choice of value  from Lemma 9. In addition, parameter p(N − 1) should be large enough (as was stated in Theorem 6).

5 Mixed Pairs of Singular N-Particle Boxes It remains to derive the property (DS.k + 1, I, N) in case (III), i.e., for mixed pairs of N-particle boxes (where one is FI and the other PI). Here we use several properties which have been established earlier in this paper for all scale lengths, namely, (WS1.n), (WS2.n) for n = 1, . . . , N, NITRoNS, and the inductive assumption (IS.k + 1.N) which we have already derived from (IS.k.N) in Section 4. A natural counterpart of Theorem 6 for mixed pairs of boxes is the following Theorem 7 ∀ given interval I ⊆ R, there exists a constant L∗1 ∈ (0, +∞) with the following property. Assume that L0  L∗1 and, for a given k  0, the property x), Λ(N) y), and (ii) ∀ (DS.k, I, N) holds (i) ∀ pair of separable PI-boxes Λ(N) Lk ( Lk ( (N) (N) pair of separable FI-boxes Λ Lk ( x), Λ Lk ( y). (N) (N) Let Λ Lk+1 (x), Λ Lk+1 (y) be a pair of separable boxes, where Λ(N) Lk+1 (x) is FI and Λ(N) Lk+1 (y) PI. Then

  −2 p(N) (N) P ∃ E ∈ I : both Λ(N) . Lk+1 (x), Λ Lk+1 (y) are (E, mk+1 )−S  Lk+1

(5.1)

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Proof of Theorem 7 Recall that the Hamiltonian H(N)(N) is decomposed as ΛLk+1 (y) in Eqs. 3.1, 3.2. Consider the following three events:   (N) B = ∃ E ∈ I : both Λ(N) (x), Λ (y) are (E, m )-S , k+1 Lk+1 Lk+1   T = Λ Lk+1 (y) is m0 -PT ,   (N) R = ∃ E ∈ I : neither Λ(N) Lk+1 (x) nor Λ Lk+1 (y) is (E, J)-CNR . Recall that by virtue of Eq. 3.4, we have P{T} 

1 −2 p(N) L 4 k+1

(5.2)

For the event R we have, by virtue of Lemma 11 and inequality (4.13), −q(N)+2

P { R }  Lk+1

;

(5.3)

as before, q(N) is the parameter from Eq. 1.13. Further, P { B }  P { T } + −2 p(N) P { B ∩ Tc }  14 Lk+1 + P { B ∩ Tc }, and we have & ' & ' & ' −q(N)+2 P B ∩ Tc  P { R } + P B ∩ Tc ∩ Rc  Lk+1 + P B ∩ Tc ∩ Rc . (N) Within the event B ∩ Tc ∩ Rc , either Λ(N) Lk+1 (x) or Λ Lk+1 (y) is E-CNR. It must

(N) be the FI-box Λ(N) Lk+1 (x). Indeed, by NITRoNS (Lemma 3), had box Λ Lk+1 (y) been both E-CNR and m0 -NPT, it would have been (E, mk+1 )-NS, which is not allowed within the event B. Thus, the box Λ(N) Lk+1 (x) must be E-CNR, but (E, mk+1 )-S:

B ∩ Tc ∩ Rc ⊂ {∃ E ∈ I : Λ(N) Lk+1 (x) is (E, mk+1 )-S and E-CNR}. However, applying Lemma 12, we see that   ∃ E ∈ I : Λ(N) Lk+1 (x) is (E, mk+1 )-S and E-CNR   ⊂ ∃ E ∈ I : K(Λ(N) Lk+1 (x); E)  J + 1 . Therefore, with the same values of parameters as in Corollary 1 (J = 2 + 1,   4),     & ' P B ∩ Tc ∩ Rc  P ∃ E ∈ I : K Λ(N) (x); E  2 + 2 Lk+1 −2 p(N)

 2L−1 k+1 Lk+1

.

(5.4)

Finally, we get, with q (N) := q(N)/α,

& ' P { B }  P { B ∩ T } + P { R } + P B ∩ Tc ∩ Rc 

1 −2 p(N) −q (N)+4 −2 p(N) −2 p(N) + 2L−1  Lk+1 , Lk+1 + Lk+1 k+1 Lk+1 2

(5.5)

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137

for sufficiently large L0 , if we can guarantee, by taking |g| large enough, that q (N) > 2 p(N) + 5. This completes the Proof of Theorem 7.   Remark The Proof of Theorem 7 practically repeats that of Theorem 5.1 from [7]; the only difference is in specification of constants in the exponents. Therefore, Theorem 4 is also proven. Acknowledgements VC thanks The Isaac Newton Institute (INI) and Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, for hospitality during visits in 2003, 2004, 2007 and 2008. YS thanks the Département de Mathématiques, Université de Reims for hospitality during visits in 2003 and 2006–2008, in particular, for a Visiting Professorship in the Spring of 2003. YS thanks IHES, Bures-sur-Yvette, and STP, Dublin Institute for Advanced Studies, for hospitality during visits in 2003–2007. YS thanks the Departments of Mathematics of Penn State University and of UC Davis, for hospitality during Visiting Professorships in the Spring of 2004, Fall of 2005 and Winter of 2008. YS thanks the Department of Physics, Princeton University and the Department of Mathematics of UC Irvine, for hospitality during visits in the Spring of 2008. YS acknowledges the support provided by the ESF Research Programme RDSES towards research trips in 2003–2006. We are grateful to the INI, where our multi-particle project was originated in 2003, during the programme Interaction and Growth in Complex Systems. Special thanks go to the organisers of the 2008 INI programme Mathematics and Physics of Anderson Localization: 50 Years After; for our project the mark became First Five Years After.

Appendix A Proof of Lemma 1 Consider two N-particle configurations x and y and introduce the following notion: we shall say that the set of positions {x j, j ∈ J }, J ⊆ {1, . . . , N}, form an R-connected cluster (or simply an R-cluster) iff the set    Λ R yj ⊂ Zd (6.1) j∈J

is connected. Otherwise, this set of particles is called R-disconnected, in which case it can be decomposed into two or more R-clusters. Now, we proceed as follows. (1) Decompose the configuration y into L-clusters (of diameter  2NL). (2) To each position yj there corresponds precisely one cluster, denoted by Γ ( j). Let Y = {Γ ( j) : j ∈ J } stand for the collection of clusters, with card Y  N. (3) Consider any of the clusters Γ ( j) ∈ Y. By definition, Γ ( j) is disjoint from all other clusters: * Γ ( j), if Γ (i) = Γ ( j), Γ ( j) ∩ Γ (i) = (6.2) ∅, otherwise. Therefore, for any two distinct clusters Γ  , Γ  ∈ Y, the respective sigmaalgebras B(Γ  ), B(Γ  ) are independent.

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(4) Suppose that ∃ j ∈ {1, . . . , N} : Γ ( j) ∩ ΠΛ(N) L (x) = ∅. Set   ¯ j(y) := B ∪Γ (i)=Γ ( j ) Γ (i) . B Then the sigma-algebra B(Γ ( j)) is independent of B(Λ(N) L (x)) and of ¯ j(y): B +  (N)  + ¯ j(y). B (Γ ( j)) B Λ L (x) , B(Γ ( j)) B (6.3) (N) In other words, the box Λ(N) L (y) is separable from Λ L (x). (5) Suppose (4) is wrong, and let’s deduce from the negation of (4) a necessary condition on possible locations of the configuration y, so as to show that the number of possible choices is finite. Indeed our hypothesis reads as follows:

∀ j ∈ {1, . . . , N} Y( j) ∩ ΠΛ(N) L (x)  = ∅.

(6.4)

Therefore, ∀ j ∈ {1, . . . , N} ∃ i : y j − xi   4NL + L = (4N + 1)L  5NL ⇒ ∀ j ∈ {1, . . . , N} y j ∈ ΠΛ(N) AL (x), A = A(N) = 5N. We see that if a configuration y is not separable from x, then every d position y j must belong to one of the boxes Πi Λ(N) AL (x) = Λ AL (xi ) ⊂ Z . The total number k of these boxes is bounded by N. There are at most k N /k! choices of the boxes Λ AL (xi ) for the N positions y1 , . . . , y N . For   k N /k!  K(N)  any given choice among J(N) = J(N, K) possibilities,  with K(N) < ∞, the point y = (y1 , . . . , y N ) must belong to the Cartesian product of N boxes of size AL, i.e. to an Nd-dimensional box of size AL. The assertion of Lemma 1 now follows.  

Appendix B: Finite-Volume Localisation Bounds Here we give the proof of Lemma 3. Recall, we consider operator H(N) ΛLk (u) (N) (N) in a box Λ Lk (u). Let Ψ j, j = 1, . . . , |Λ Lk |, be its normalised EFs and E j the respective EVs. Fix j and consider the GFs G(N) (v, y; E j), v, y ∈ Λ(N) . Proof of Lemma 3 Recall that the CNR property implies NR. Observe that E − λa − μb = (E − λa ) − μb . Further, by the hypothesis of the lemma, (n )   Λ(N) Lk (u) is E-CNR. Therefore, for all λa , the n -particle box Λ Lk (u ) is 

)  (E − λa )-NR. By the assumption of m-NPT, ∀ E ∈ I box Λ(n Lk (u ) must not contain two disjoint (E − λa , m)-S sub-boxes of size Lk−1 . Therefore, the MSA  )  procedure proves that Λ(n Lk (u ) is (E − λa )-NS, yielding the required upper bound.

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139

Let us now prove the second assertion of the lemma. If v = (v , v ) ∈     ∂Λ(N) Lk (u), then either u − v  = Lk , or u − v  = Lk . In the former case we can write           ψb u ψb v (N)   G (u, v; E) = ϕa u ϕa v (E − λa ) − μb a b      (n )     = ϕa u ϕa v G (n ) u , v ; E − λa . (7.1)  ΛLk (u ) a Since ϕa  = 1, we see that    ) (n )  |G(N) (u, v; E) |  Λ(n Lk u max |G (n )

    u , v ; E − λa |.

(7.2)

In the case where u − v  = L, we can use the representation         G(N) (u, v; E) = ψb u ψb v G(n(n) )  u , v ; E − μb .

(7.3)

λa

b

Λ L (u ) k

Λ L (u ) k

  Now, as was said before, Lemma 4 follows from Lemma 3 combined with the bounds (DS.k, I, n ), (DS.k, I, n ), for 1  n , n < N.

References 1. Aizenman, M., Molchanov, S.: Localization at a large disorder and at extreme energies: an elementary derivation. Comm. Math. Phys. 157, 245–278 (1993) 2. Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractionalmoment criteria for Anderson localization. Comm. Math. Phys. 224, 219–253 (2001) 3. Aizenman, M., Warzel, S.: Localization bounds for multiparticle systems. arXiv:0809:3436 (2008) 4. Chulaevsky, V.: A Wegner-type estimate for correlated potentials. Math. Phys. Anal. Geom. 11, 117–129 (2008) 5. Chulaevsky, V., Suhov, Y.: Wegner-Stollmann bounds and localization in correlated potentials. Université de Reims. http://helios.univ-reims.fr/Labos/UMR6056/VC/WS-corrBeamer.pdf (2007) 6. Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283, 479–489 (2008) 7. Chulaevsky, V., Suhov, Y.: Eigenfunctions in a two-particle Anderson tight binding model. Comm. Math. Phys. (2009, in press). doi:10.1007/s00220-008-0721-0 8. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Comm. Math. Phys. 124, 285–299 (1989) 9. Fröhlich, J., Spencer, T.: Absence of diffusion inthe Anderson tight binding model for large disorder or low energy. Comm. Math. Phys. 88, 151–184 (1983) 10. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization in Anderson tight binding model. Comm. Math. Phys. 101, 21–46 (1985) 11. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. arXiv:0704:2664 (2007) 12. Stollmann, P.: Wegner estimates and localization for continuous Anderson models with some singular distributions. Arch. Math. 75, 307–311 (2000) 13. Stollmann, P.: Caught by Disorder. Birkhäuser, Boston (2001)

Math Phys Anal Geom (2009) 12:141–156 DOI 10.1007/s11040-009-9056-0

On Ising Model with Four Competing Interactions on Cayley Tree N. N. Ganikhodjaev · U. A. Rozikov

Received: 15 July 2008 / Accepted: 21 January 2009 / Published online: 10 February 2009 © Springer Science + Business Media B.V. 2009

Abstract In the paper we consider an Ising model with four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) on the Cayley tree of order two. We show that for some parameter values of the model there is a phase transition. Our second result gives complete description of the periodic Gibbs measures for the model. We also construct uncountably many non-periodic extreme Gibbs measures. Keywords Cayley tree · Ising model · Competing interactions · Phase transition · Gibbs measure Mathematics Subject Classifications (2000) 60K35 · 82B20 · 82B05

1 Introduction Lattice spin systems are a large class of systems considered in statistical mechanics. Some of them have a real physical meaning, others are studied as suitably simplified models of more complicated systems. The structure of the

N. N. Ganikhodjaev International Islamic University Malaysia, P.O. Box 141, 25710, Kuantan, Malaysia e-mail: [email protected] N. N. Ganikhodjaev National University of Uzbekistan, Vuzgorodok, 100095, Tashkent, Uzbekistan U. A. Rozikov (B) Institute of Mathematics and Information Technologies, 29, F. Hodjaev str., Tashkent, 100125, Uzbekistan e-mail: [email protected]

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lattice plays an important role in investigations of spin systems. For example, in order to study the phase transition problem for a system on Z d and on Cayley tree there are two different methods: Pirogov-Sinai theory on Z d , Markov random field theory and recurrent equations of this theory on Cayley tree. In [2–6] for several models on Cayley tree, using the Markov random field theory Gibbs measures are described. In the paper we investigate a model with four competing interactions on the Cayley tree. The paper is organized as follows. In Section 2 we give definitions of the model, Cayley tree and Gibbs measures. In Section 3 we reduce the problem of describing limit Gibbs measures to the problem of solving a nonlinear functional equations. Section 4 devoted to describe translation-invariant Gibbs measures. We show that two (minimal and maximal) of translation-invariant Gibbs measures are extreme in the set of all Gibbs measures. In Section 5 we study periodic Gibbs measures and show that our model admits only translation-invariant and periodic with period two (chess-board) Gibbs measures. In the last section we construct uncountably many non-periodic extreme Gibbs measures.

2 Definitions Cayley tree The Cayley tree  k (see [1]) of order k  1 is an infinite tree, i.e. a graph without cycles, from each vertex of which exactly k + 1 edges issue. Let  k = (V, L, i) where V is the set of vertices of  k , L is the set of edges of  k and i is the incidence function associating each edge l ∈ L with its endpoints x, y ∈ V. If i(l) = {x, y}, then x and y are called nearest neighboring vertices and we write l =< x, y >. The distance d(x, y), x, y ∈ V on the Cayley tree is defined by the formula d(x, y) = min{d|x = x0 , x1 , ..., xd−1 , xd = y ∈ V such that the pairs < x0 , x1 >, ..., < xd−1 , xd > are neighboring vertices}. For the fixed x0 ∈ V we set     Wn = x ∈ V|d x, x0 = n ,

    Vn = x ∈ V|d x, x0  n ,

Ln = {l =< x, y >∈ L|x, y ∈ Vn }. A collection of the pairs < x, x1 >, ..., < xd−1 , y > is called a path from x to y. We write x < y if the path from x0 to y goes through x. We call the vertex y a direct successor of x, if y > x and x, y are nearest neighbors. The set of the direct successors of x is denoted by S(x), i.e. S(x) = {y ∈ Wn+1 |d(x, y) = 1},

x ∈ Wn .

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We observe that for any vertex x = x0 , x has k direct successors and x0 has k + 1. The vertices x and y are called second neighbor which is denoted by > x, y <, if there exists a vertex z ∈ V such that x, z and y, z are nearest neighbors. We will consider only second neighbors > x, y <, for which there exist n such that x, y ∈ Wn . Three vertices x, y and z are called a triple of neighbors and they are denoted by < x, y, z >, if < x, y >, < y, z > are nearest neighbors and x, z ∈ Wn , y ∈ Wn−1 , for some n ∈ N. The fixed vertex x0 is called the 0-th level and the vertices in Wn are called the n-th level. It is known [5] that there exists a one-to-one correspondence between the set V of the vertices of the Cayley tree of order k  1 and the group Gk of the free products of k + 1 cyclic groups of the second order with generators a1 , a2 , ..., ak+1 . Let us define a group structure on the  k as follows. Vertices which corresponds to the ‘words’ g, h ∈ Gk are called nearest neighbors and are connected by an edge if either g = hai or h = ga j for some i or j. The graph thus defined is a Cayley tree of order k. Consider a left (resp. right) transformation shift on Gk defined as : for go ∈ Gk we put Tg0 h = g0 h(resp.Tg0 h = g0 h) ∀h ∈ Gk . Then the set of all left (resp. right) shifts on Gk is isomorphic to the group Gk . The model The Ising model, which was originally regarded as a ferromagnetic model, has found some applications in many other physical, biological and chemical systems, and even in sociology. The model that considered in [8] is a natural generalization of the Ising model, and a model of the similar form has recently been investigated by Monroe [15, 16] to understand the physical aspects associated with the Husimi tree or the Kagome lattice. On a similar note, the topic of statistical mechanics on non amenable graphs is a modern growing field [2, 14]. In the same paper [8], we have presented the exact solution of an Ising model with competing restricted interactions and zero external magnetic field on the Cayley tree  2 for order 2. In this paper we consider the Ising Model with four competing interactions on the Cayley tree which is defined by the following Hamiltonian   H(σ ) = −J3 σ (x)σ (y)σ (z) − J σ (x)σ (y) − <x,y,z>

−J1



<x,y>

σ (x)σ (y) − α



>x,y<

σ (x)

(1)

x∈V

where the sum in the first term ranges all triples of neighbors, the second sum ranges all second neighbors, the third sum ranges all nearest neighbors and the spin variables σ (x) assume the values ±1. (See [1] for models with competing interactions, and see [2, 14–16] for the physical motivation underlying the study of these models.)

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Remark If one consider the Hamiltonian with all possible triple (without condition x, z ∈ Wn ) and second neighbors (without condition x, y ∈ Wn ) then the problem of describing of limit Gibbs measures becomes a difficult problem. Note that interactions defined in (1) depend on the chosen origin x0 , another choice of x0 would give different interactions. Thus the tree-spin interactions are not translation invariant. The various partial cases of this model have been investigated in numerous works, for example, the case J3 = α = 0 was considered in [15, 16] and [8]. In [8], the exact solution of an Ising model with competing restricted interactions with zero external field was presented. The case J = α = 0 was considered in [7, 16], and [17]. In [7], the exact solution was found for the problem of phase transitions. In [17] it is proven that there are two translation—invariant and uncountable number of distinct non-translation—invariant extreme Gibbs measures. In [9] the phase transition problem was solved for α = 0, J · J1 · J3 = 0 and for J3 = 0, α · J · J1 = 0 as well. In the paper we will consider the case J · J1 · J3 · α = 0. Gibbs measures Let  be a finite subset of V. Assume () is the set of all configuration on , that is the functions {σ (x), x ∈ }. Let σ (V \ ) be a fixed boundary configuration. The total energy of configuration σ () ∈ () under condition σ (V \ ) is defined as H(σ ()|σ (V \ )) = −J3



σ (x)σ (y)σ (z) − J

< x, y, z > x, y, z ∈ 

− J1



σ (x)σ (y) − α

< x, y > x, y ∈ 



> x, y < x ∈ , y ∈ /

σ (x) −

σ (x)σ (y)σ (z) −

< x, y, z > x ∈ , y ∈ / , z ∈ /  or x ∈ , y ∈ , z ∈ /

−J

σ (x)σ (y) −

> x, y < x, y ∈ 

x∈



− J3





σ (x)σ (y) − J1



σ (x)σ (y) .

(2)

< x, y > x ∈ , y ∈ /

When all boundary points {σ (y), y ∈ V \ } are fixed as +1, we have the positive boundary condition and when they are fixed as −1, we have negative boundary condition. The free boundary condition corresponds to the case when the last three sums in the above are absent, that is formally all boundary points are fixed as 0. The partition function Z  (σ (V \ )) in volume  under boundary condition σ (V \ )) is defined as Z =

 σ ()∈()

exp(−β H(σ ())|σ (V \ )),

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where β = 1/kT is the inverse temperature. Then the conditional Gibbs measure μ in volume  under boundary condition σ (V \ ) is defined as μ (σ ()) =

exp(−β H(σ ())|σ (V \ )) . Z

(3)

3 The Functional Equation There are several approaches to derive the equation solutions of which describes the limit Gibbs measures for lattice models on the Cayley tree. One approach is based on properties of Markov random fields on Cayley tree [23] and [18]. Another approach is based on recurrence equations for partition functions [7, 12]. Here we shall use the Markov random field method. Let h : x → R be a real valued function of x ∈ V. Given n = 1, 2, ..., consider the probability measure μ(n) on {−1, +1}Vn which defined by    (n) −1 hx σ (x) . μ (σn ) = Z n exp − β H(σn ) + x∈Wn

Here, as before, β = 1/kT and σn : x ∈ Vn → σn (x) and Z n is the corresponding partition function     Zn = exp − β H(σ˜ n ) + hx σ˜ (x) . σ˜ n ∈(Vn )

x∈Wn

The consistency condition for μ(n) (σn ), n  1 is    μ(n) σn−1 , σ (n) = μ(n−1) (σn−1 ),

(4)

σ (n)

where σ (n) = {σ (x), x ∈ Wn }. Let V1 ⊂ V2 ⊂ ..., ∪∞ n=1 Vn = V and μ1 , μ2 , ... be a sequence of the probability measures on V1 , V2 , ... satisfying the consistency condition, where  = {−1, +1}. Then, according to the Kolmogorov theorem, (see, e.g. [21]), there is a unique limit Gibbs measure μh on  such that for every n = 1, 2, ... and σn ∈ Vn the following equality holds μ({σ |Vn = σn }) = μ(n) (σn ). The following statement describes the conditions on hx which guarantee the consistency condition of measures μ(n) (σn ). Proposition 1 The measure μ(n) (σn ), n = 1, 2, ... satisfies the consistency condition (4) if and only if for any x ∈ V the following equation holds:   2

θ1 θ2 θ3 e2(h y +hz ) + θ1 e2h y + e2hz + θ2 θ3 1  2h  hx = log θ4 2 , (5) 2 θ1 θ2 + θ1 θ3 e y + e2hz + θ2 e2(h y +hz )

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here S(x) = {y, z}, < y, x, z > is a ternary neighbor and θ1 = e2β J1 , θ2 = e2β J , θ3 = e2β J3 , θ4 = e2βα . Proof Necessity. According to the consistency condition (4) we have   Z n−1  exp − β Hn−1 (σn−1 ) + β J1 σ (x)(σ (y) + σ (z)) + Z n (n) x∈W , σ



+ βJ

n−1 y, z ∈ S(x)

σ (y)σ (z) + β J3

x ∈ Wn−1 , y, z ∈ S(x)

+

σ (x)σ (y)σ (z) +

x ∈ Wn−1 , y, z ∈ S(x)



βασ (x) +

x∈Wn−1

 = exp





 hx σ (x)

x∈Wn−1

− β Hn−1 (σn−1 ) +



 hx σ (x)

x∈Wn−1

Consequently we have Z n−1   exp{β J1 σ (x)(σ (y) + σ (z)) + β Jσ (y)σ (z) + Z n (n) x∈W σ

n−1

+ β J3 σ (x)σ (y)σ (z) + βασ (x) + h y σ (y) + hz σ (z)}



=

exp{hx σ (x)}.

x∈Wn−1

Assume x ∈ Wn−1 and S(x) = {y, z}, σx(n) = {σ (y), σ (z)}. Since σ (n) = ∪x∈Wn−1 σx(n) , we get Z n−1   exp{β J1 σ (x)(σ (y) + σ (z)) + β Jσ (y)σ (z) + Z n x∈W (n) n−1

=



σx

+ β J3 σ (x)σ (y)σ (z) + βασ (x) + h y σ (y) + hz σ (z)} exp{hx σ (x)}.

(6)

x∈Wn

Now fix x ∈ Wn−1 and rewrite (6) for the cases σ (x) = 1 and σ (x) = −1. If σ (x) = 1, we have  N= exp{β J1 (σ (y) + σ (z)) + β Jσ (y)σ (z) + σx(n) ={σ (y),σ (z)}

+ β J3 σ (y)σ (z) + βα + h y σ (y) + hz σ (z)} = exp{hx };

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and if σ (x) = −1, then  D=

147

exp{−β J1 (σ (y) + σ (z)) + β Jσ (y)σ (z)} +

σx(n) ={σ (y),σ (z)}

+ β J3 σ (y)σ (z) − βα + h y σ (y) + hz σ (z)} = exp{−hx }. So that N = exp{2hx }. D

(7)

The numerator N of the left-hand side is equal to N = exp(2β J1 + β J + β J3 + βα + h y + hz ) + + exp(−β J − β J3 + βα − h y + hz ) + exp(−β J − β J3 + βα + h y − hz ) + + exp(−2β J1 + β J + β J3 + βα − h y − hz ), while the denumerator D is equal to D = exp(−2β J1 + β J + β J3 − βα + h y + hz ) + + exp(−β J − β J3 − βα − h y + hz ) + exp(−β J − β J3 − βα + h y − hz ) + + exp(2β J1 + β J + β J3 − βα − h y − hz ). Then the equality N = exp{2hx } implies (5). D Sufficiency. Assume that (5) is valid. From (7) we get  exp{β J1 σ (x)(σ (y) + σ (z)) + β Jσ (y)σ (z) + σx(n) ={σ (y),σ (z)}

+ β J3 σ (x)σ (y)σ (z) + βασ (x) + h y σ (y) + hz σ (z)} = a(x) exp{σ (x)hx }, where σ (x) = ±1. This equality implies   exp{β J1 σ (x)(σ (y) + σ (z)) + β Jσ (y)σ (z) + x∈Wn−1 σx(n) ={σ (y),σ (z)}

=



+ β J3 σ (x)σ (y)σ (z) + βασ (x) + h y σ (y) + hz σ (z)} a(x) exp{σ (x)hx }.

(8)

x∈Wn−1

Denoting An (x) =

x∈Wn−1

a(x), we have from (8)

Z n−1 An−1 μ(n−1) (σn−1 ) = Z n

 σ (n)

  μ(n) σn−1 , σ (n) .

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As μ(n) , n  1 is a probability, we have     (n−1) μ(n) σn−1 , σ (n) = μ (σn−1 ) = 1. σn−1 σ (n)

σn−1

From these equalities we get Z n−1 An−1 = Z n , which means that (4) holds. According to Proposition 1 the problem of describing the Gibbs measures is reduced to the description of the solutions of the functional Eq. 5. Denote  = {−1, +1}V . Note that any transformation S of the group Gk induces a shift automorphism S˜ :  →  by ˜ )(g) = σ (Sg), ( Sσ

g ∈ Gk , σ ∈ .

By Gk we denote the set of all shifts on . We say that a Gibbs measure μ on  is translation - invariant if for any T ∈ Gk the equality μ(T(A)) = μ(A) is valid for all A ∈ F , where F is a standard σ -algebra of subsets of  generated by cylinder subsets.

4 Translation-invariant Gibbs Measures: Phase Transition The analysis of the solution of (5) is rather tricky. It is natural to begin with the translation-invariant solutions where hx = h is constant for all x ∈ V. In this case from (5), we have u = θ4

θ12 θ2 θ3 u2 + 2θ1 u + θ2 θ3 θ12 θ2 + 2θ1 θ3 u + θ2 u2

(9)

where u = e2h . Note that if there is more than one positive solution for Eq. 9, then there is more than one translation-invariant Gibbs measure corresponding to these solutions. We say that a phase transition occurs for model (1), if Eq. 9 has more than one positive solution. The number of the solutions of Eq. 9 depends on the 1 . The phase transition usually occurs for low temperature. parameter β = kT If it is possible to find an exact value of temperature T ∗ such that a phase transition occurs for all T < T ∗ then T ∗ is called a critical value of temperature. Finding the exact value of the critical temperature for some models means to exactly solve the models. Proposition 2 If θ12 > 3 , θ2 >

2θ1 , θ12 −3

θ22 (θ14 + 2θ12 − 3) − 4θ12 − 8θ1 θ2 −

<

and θ22 (θ14 + 2θ12 − 3) − 4θ12 − 8θ1 θ2 − 4θ12 θ22

2θ1 θ2 θ22 (θ14 + 2θ12 − 3) − 4θ12 − 8θ1 θ2 +

θ22 (θ14 + 2θ12 − 3) − 4θ12 − 8θ1 θ2 − 4θ12 θ22

2θ1 θ2 η1 (θ1 , θ2 , θ3 ) < θ42 < η2 (θ1 , θ2 , θ3 )

< θ3 < ,

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then Eq. 9 has three positive roots u∗1 < u∗2 < u∗3 . Here ηi (θ1 , θ2 , θ3 ) =

1 θ12 θ2 θ3 ui2 + 2θ1 ui + θ2 θ3 ui θ12 θ2 + 2θ1 θ3 ui + θ2 ui2

where ui , i = 1, 2 are the solutions of   θ12 θ22 θ3 u4 + 4θ1 θ2 u3 + θ3 3θ22 − θ14 θ22 + 4θ12 u2 + 4θ1 θ2 θ32 u + θ12 θ22 θ3 = 0. (10) Proof Denote f (u) =

θ12 θ2 θ3 u2 + 2θ1 u + θ2 θ3 . θ12 θ2 + 2θ1 θ3 u + θ2 u2

We have

      θ1 θ12 θ32 − 1 u2 + θ2 θ3 θ14 − 1 u + θ1 θ12 − θ32 , f (u) = 2θ2  2 2 θ1 θ2 + 2θ1 θ3 u + θ2 u2  −3 f (u) = 2θ2 θ2 u2 + 2θ1 θ3 u + θ12 θ2 ×        × − 2θ1 θ2 θ12 θ32 − 1 u3 − 3θ22 θ3 θ14 − 1 u2 + 6θ1 θ2 θ32 − θ12 u+     + θ12 θ3 θ22 θ24 − 1 − 4θ12 + 4θ32 .

Denote

  A = 2θ1 θ2 θ12 θ32 − 1 ;   C = 6θ1 θ2 θ32 − θ12 ;

  B = 3θ22 θ3 θ14 − 1 ;     D = θ12 θ3 θ22 θ24 − 1 − 4θ12 + 4θ32 .

It is easy to see that under conditions of the proposition we have A > 0, B > 0, C > 0, D > 0. Equation f (u) = 0 is equivalent to Au3 + Bu2 − Cu − D = 0, one can easily prove that the last equation has unique positive solution, say u∗ . Thus f is convex for u < u∗ and concave for u > u∗ . Consequently there are at most three solutions. On the other hand, it is easy to see that (9) has more than one solution if and only if there is more than one solution of the equation u f (u) = f (u) which is equivalent to Eq. 10. Now consider (10), which can be rewritten as



1 2 u θ3 θ12 θ22 u + + 4θ1 θ2 + + 3θ22 − θ14 θ22 + 4θ12 − 2θ12 θ22 = 0. u θ3 u Denote

ϕ(u) = 4θ1 θ2 ψ(u) =

θ22



θ14

u θ3 + θ3 u

+

2θ12

, 

−3 −

θ12 θ22

1 u+ u

2 − 4θ12 .

A simple analysis of these functions show that under conditions of the proposition the Eq. 9 has three positive solutions. This completes the proof.



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Thus by Propositions 1 and 2 we can formulate the following Theorem 3 Assume the conditions of the Proposition 2 are satisfied then for the model (1) there are three translation-invariant Gibbs measures μ1 , μ2 , μ3 i.e. there is phase transition. Note that μ1 (μ3 ) corresponds to positive (resp. negative) boundary condition. The boundary condition corresponding to μ2 unclear. The following Proposition 4 describes a useful property of general (non translation-invariant) solutions hx to (5). Proposition 4 Assume the conditions of the Proposition 2 are satisfied and hx is a solution of (5), with ux = e2hx , then u∗1  ux  u∗3 ,

x∈V

(11)

where u∗1 < u∗3 are solutions of (9). Proof It is clear that ux > 0, for any x ∈ V. For u, v > 0 denote F(u, v) = θ4

θ12 θ2 θ3 uv + θ1 (u + v) + θ2 θ3 . θ12 θ2 + θ1 θ3 (u + v) + θ2 uv

The Eq. 5 can be rewritten as ux = F(u y , uz ). Observe that under conditions of the Proposition 2 the function F(u, v) is increasing with respect to u and v on (0, ∞). Hence we conclude that θ3 θ4 < F(u, v) < θ12 θ3 θ4 , θ12 for all u, v > 0. Now we consider the function with u, v ∈ similar reason as above we get

  θ3 θ4 f < F(u, v) < f θ12 θ3 θ4 , 2 θ1

 θ3 θ4 θ12

 , θ12 θ2 θ3 . By

where f (u) = F(u, u). Repeating this argument one gets

  θ3 θ4 f (n) < F(u, v) < f (n) θ12 θ3 θ4 , 2 θ1 for all n  1. Here f (n) is n-th iteration of the map x → f (x). The sequence f (n) (θ12 θ3 θ4 ) is decreasing and bounded below by u∗3 . Its limit is a fixed point of f and thus equal to u∗3 . This proves that ux  u∗3 . The lower bound for ux is

similar and gives u∗1 .

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Using Proposition 4 by similar argument as in the proof of Theorem 12.31 of [10] one can prove the following Theorem 5 Assume conditions of Proposition 2 are satisfied then translationinvariant measures μ1 , μ3 (see Theorem 3) are extreme. Remark The problem of extremality for measure μ2 is a difficult problem. Usually (see [3, 24]) such measure which corresponds to unordered phase is extreme for the temperature T ∈ (Tc , Tc ] where Tc is the critical temperature of phase transition and Tc is a (second) critical temperature (0 < Tc < Tc ).

5 Periodic Gibbs Measures In this section we study a periodic (see Definition 6) solutions of (5). Definition 6 Let K be a subgroup of Gk , k  1. We say that a collection (of functions) h = {hx ∈ R1 : x ∈ Gk } is K-periodic if h yx = hx for all x ∈ Gk and y ∈ K. Definition 7 A Gibbs measure is called K-periodic if it corresponds to Kperiodic collection h. Observe that a translation-invariant Gibbs measure is Gk -periodic. We give a complete description of periodic Gibbs measures i.e. a characterization of such measures with respect to any normal subgroup of finite index in Gk . Let K be a subgroup of index r in Gk , and let Gk /K = {K0 , K1 , ..., Kr−1 } be the quotient group, with the coset K0 = K. Let qi (x) = |S1 (x) ∩ Ki |, i = 0, 1, ..., r − 1; N(x) = |{ j : q j(x) = 0}|, where S1 (x) = {y ∈ Gk : x, y}, x ∈ Gk and | · | is the number of elements in the set. Denote Q(x) = (q0 (x), q1 (x), ..., qr−1 (x)). We note (see [19]) that for every x ∈ Gk there is a permutation πx of the coordinates of the vector Q(e) (where e is the identity of Gk ) such that πx Q(e) = Q(x).

(12)

It follows from (12) that N(x) = N(e) for all x ∈ Gk . Each K− periodic collection is given by {hx = hi for x ∈ Ki , i = 0, 1, ..., r − 1}. For k = 2 by Proposition 1 and (12), hn , n = 0, 1, ..., r − 1, satisfies hn =

  1 log F e2hπn (i) , e2hπn ( j) , 2

(13)

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where F(u, v) is defined in proof of the Proposition 4 and πn is permutation of Q(e) for x ∈ Kn , i, j ∈ Q(e). Proposition 8 Suppose the conditions of Proposition 2 are satisfied then F(u, v) = F(h, v) if and only if u = h (F(u, v) = F(u, h) if and only if v = h). Proof Follows from monotonity of F with respect to u (resp. v).



Let G∗2 be the subgroup in G2 consisting of all words of even length. Clearly, is a subgroup of index 2.

G∗2

Theorem 9 Let K be a normal subgroup of finite index in G2 . Then each K− periodic Gibbs measure for model (1) is either translation-invariant or G∗2 − periodic. Proof We see from (13) that     F ehπn (i) , ehπn ( j) = F ehπn (i ) , ehπn ( j ) ,

(14)

For any i, j, i , j ∈ Q(e), n = 0, 1, ..., r − 1. Hence from Proposition 8 we have hπn (i1 ) = hπn (i2 ) = ... = hπn (i N(e) ) . Therefore, hx = h y = h, hx = h y = l,

if x, y ∈ S1 (z),

if x, y ∈ S1 (z),

z ∈ G∗2 ; z ∈ G2 \ G∗2 .

Thus the measures are translation-invariant (if h = l) or G∗2 − periodic (if h = l). This completes the proof of the theorem.

Let K be a normal subgroup of finite index in G2 . What condition on K will guarantee that each K−periodic Gibbs measure is translation-invariant? We put I(K) = K ∩ {a1 , a2 , a3 }, where ai , i = 1, 2, 3 are generators of G2 . Theorem 10 If I(K) = ∅, then each K− periodic Gibbs measure for model (1) is translation-invariant. Proof Take x ∈ K. We note that the inclusion xai ∈ K holds if and only if ai ∈ K. Since I(K) = ∅, there is an element ai ∈ K. Therefore K contains the subset Kai = {xai : x ∈ K}. By Theorem 9 we have hx = h and hxai = l. Since x and xai belong to K, it follows that hx = hxai = h = l. Thus each K− periodic Gibbs measure is translation-invariant. This proves Theorem 10.

Theorems 9 and 10 reduce the problem of describing K− periodic Gibbs measure with I(K) = ∅ to describing the fixed points of f (u) = F(u, u) (see

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(9)) which describes translation-invariant Gibbs measures. If I(K) = ∅, this problem is reduced to describing the solutions of the system:  u = f (v), (15) v = f (u) with f (u) = θ4

θ12 θ2 θ3 u2 + 2θ1 u + θ2 θ3 . θ12 θ2 + 2θ1 θ3 u + θ2 u2

Evidently the positive roots of the equation f ( f (u)) − u =0 f (u) − u

(16)

describe the periodic (non translation-invariant) Gibbs measures. As we are looking for positive roots (16) has the following form:     θ12 θ2 θ12 θ2 θ32 θ42 +2θ1 θ32 θ4 +θ2 u2 + θ3 θ14 θ22 θ4 +2θ13 θ2 θ42 +2θ12 θ2 +4θ12 θ4 − θ22 θ4 u +   + θ12 θ2 θ2 θ32 θ42 + 2θ1 θ4 + θ12 θ2 = 0, (17) The discriminant  of (17) is equal to  = −4θ15 θ23 θ43 (θ1 θ2 θ4 + 2)θ34 + Aθ32 − 4θ15 θ23 (θ1 θ2 + 2θ4 ), where

     A = − θ42 3θ18 + 6θ14 − 1 θ24 − 4θ13 θ4 1 + θ42 1 + θ14 θ23 +     + 4θ12 θ14 θ44 + θ14 − 2θ42 θ22 + 16θ15 θ4 1 + θ42 θ2 + 16θ14 θ42 .

Using simple analysis one can see that (17) has two positive solutions if θ1 < 1, θ2 >

2θ1 , 1 − θ12

where θ4∗ =

θ22 − 4θ12 − θ14 θ22 +

1 < θ4 < θ4∗ , θ4∗

(18)

 2 4θ12 + θ14 θ22 − θ22 − 16θ16 θ22 4θ13 θ2

and A2 > 64θ110 θ26 θ4 (θ1 θ2 θ4 + 2)(θ1 θ2 + 2θ4 ), θ3−

<

θ32

<

θ3+ ,

(19) (20)

where θ3∓ are solutions of  = 0. Therefore, the following theorem is proved: Theorem 11 Assume (θ1 , θ2 , θ3 , θ4 ) satisfied conditions (18)–(20) then for the per per model (1) there are two G∗2 − periodic Gibbs measures μ1 , μ2 .

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Remark per

per

1. By construction measures μ1 , μ2 are non translation-invariant, but periodic with period 2 (= index of normal subgroup). 2. For θ4 = 1 the condition (19) can be rewritten as (see [9])



2

  2  2 2θ1 2θ1 2θ1 2 2 2 1 − 3θ1 θ1 + 1 θ2 − θ2 − θ2 + > 0. 1 − 3θ12 1 + θ12 1 + θ12 This factorization gives more simple formulation of the conditions (18)–(20) i.e. for θ4 = 1 conditions (18)–(20) can be reduced to 1 0 < θ1 < √ , 3

θ2 >

2θ1 , 1 − 3θ12

θ3− < θ3 < θ3+ .

6 Non Periodic Gibbs Measures In this section we consider the case of phase transition (i.e. assume that the conditions of Proposition 2 are satisfied). We show that functional Eq. 5 admits uncountably many non periodic solutions. Take an arbitrary infinite path π = {x0 = x0 , x1 , ...} on the Cayley tree of order 2. There is (see [4, 20]) one-to-one correspondence between such paths and real numbers t ∈ [0; 1]. We will map the path π to a function hπ : x ∈ V → hπx satisfying (5). Path π splits Cayley tree  2 into two parts 12 and 22 . Function hπ is defined by  log u∗1 , if x ∈ 12 π hx = (21) log u∗3 , if x ∈ 22 Denote (x, y) =

 θ 2 θ θ e2(x+y) + θ (e2x + e2y ) + θ θ  1 2 3 1 2 3 log θ4 1 2 2x 2y 2(x+y) 2 θ1 θ2 + θ1 θ3 (e + e ) + θ2 e

Proposition 12 Following inequality holds: |(x1 , y) − (x2 , y)|  γ (θ1 , θ2 , θ3 )|x1 − x2 |, where

√ √ | (θ1 t + θ2 θ3 )(θ2 t + θ1 θ3 )−θ1 (θ1 θ2 θ3 t + 1)(θ3 t + θ1 θ2 )| γ (θ1 , θ2 , θ3 ) = max <1 √ √ t∈[u∗1 ,u∗3 ] (θ1 t + θ2 θ3 )(θ2 t + θ1 θ3 )+θ1 (θ1 θ2 θ3 t + 1)(θ3 t + θ1 θ2 )

Proof The function (x, y) can be rewritten as (x, y) =

Ae2x + B 1 1 log θ4 + log 2x , 2 2 Ce + D

On Ising Model with Four Competing Interactions on Cayley Tree

155

where A, B, C, D depends on θ1 , θ2 , θ3 and y. It is easy to see that √ √ | AD − BC| . |x (x, y)|  √ √ AD + BC This completes the proof.



With the help of Proposition 12 it is easy to prove the following Theorem 13, similar to Theorem 3 of [20]: Theorem 13 For any infinite path π , there exists a unique function hπ satisfying (5) and (21). In the standard way (see e.g. [4, 20]) one can prove that functions hπ(t) are different for different t ∈ [0; 1]. Now let μ(t) denote the Gibbs measure corresponding to function hπ(t) , t ∈ [0; 1]. Using Theorem 5, similar to analogous theorem of [4] we obtain the following Theorem 14 For any t ∈ [0; 1], there exists a unique extreme Gibbs measure μ(t). Moreover, the above Gibbs measures μi , i = 1, 3, are specified as μ(0) = μ3 , μ(1) = μ1 . Because measures μ(t) are different for different t ∈ [0; 1] we obtain a continuum of distinct extreme Gibbs measures. Acknowledgements The final part of this work was done in the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy and authors thank ICTP for providing financial support and all facilities. We also thank IMU/CDE-program for travel support. The work particularly supported (first author) by the IIUM short term research Grant and (second author) by NATO Reintegration Grant: FEL.RIG. 980771.

References 1. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982) 2. Bleher, P., Ruiz, J., Schonmann, R.H., Schlosman, S., Zagrebnov, V.: Rigidity of the critical phases on a Cayley tree. Moscow Math. J. 1, 345–363 (2001) 3. Bleher, P., Ruiz, J., Zagrebnov, V.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79, 473–482 (1995) 4. Bleher, P., Ganikhodjaev, N.: On the phases of the Ising model on the Bethe lattice. Theory Probab. Appl. 35, 216–227 (1990) 5. Ganikhodjaev, N.N.: Group representations and automophisms of the Cayley tree. Dokl. Akad. Nauk. Rep. Uzbekistan 4, 3–5 (1994) (Russian) 6. Ganikhodjaev (Ganikhodzhaev), N.N., Rozikov, U.A.: A description of periodic extremal Gibbs measures on some lattice models on the Cayley tree. Theor. Math. Phys. 111, 480–486 (1997) 7. Ganikhodjaev, N.N.: Exact solution of an Ising model on the Cayley tree with competing ternary and binary interactions. Theor. Math. Phys. 130(3), 419–424 (2002)

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8. Ganikhodjaev, N.N., Pah, C.H., Wahiddin, M.R.B.: Exact solution of an Ising model with competing interactions on a Cayley tree. J. Phys. A 36, 4283–4289 (2003) 9. Ganikhodjaev, N.N., Pah, C.H., Wahiddin, M.R.B.: An Ising model with three competing interactions on a Cayley tree. J. Math. Phys. 45, 3645–3658 (2004) 10. Georgii, H.-O.: Gibbs Measures and Phase Transitions. Walte de Gruyter, Berlin (1998) 11. Katsura, S., Takizawa, M.: Bethe lattice and the Bethe approximation. Prog. Theor. Phys. 51, 82–98 (1974) 12. Kindermann, R., Snell, J.L.: Markov random fields and their applications. Contemp. Math. 1 (1980) 13. Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1961) 14. Lyons, R.: Phase transitions on nonamenable group. J. Math. Phys. 41, 1099–1126 (2000) 15. Monroe, J.L.: Phase diagrams of Ising models on Husime trees II. J. Stat. Phys. 67, 1185–2000 (1992) 16. Monroe, J.L.: A new criterion for the location of phase transitions for spin system on a recursive lattice. Phys. Lett. A 188, 80–84 (1994) 17. Mukhamedov, F.M., Rozikov, U.A.: On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras. J. Statist. Phys. 114, 825– 848 (2004) 18. Preston, K.: Gibbs States on Countable Sets. Cambridge, London (1974) 19. Rozikov, U.A.: Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to description of periodic Gibbs distributions. Theor. Math. Phys. 112, 929–933 (1997) 20. Rozikov, U.A.: Description uncountable number of Gibbs measures for inhomogeneous Ising model. Theor. Math. Phys. 118, 95–104 (1999) 21. Shiryaev, A.N.: Probability. Nauka, Moscow (1980) 22. Sinai, Y.G.: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982) 23. Spitzer, F.: Markov random field on infinite tree. Ann. Probab. 3, 387–398 (1975) 24. Suhov, Y.M., Rozikov, U.A.: A hard-core model on a Cayley tree: an example of a loss network. Queueing Systems 46, 197–212 (2004)

Math Phys Anal Geom (2009) 12:157–180 DOI 10.1007/s11040-009-9057-z

Degree Complexity of a Family of Birational Maps: II. Exceptional Cases Tuyen Trung Truong

Received: 29 November 2008 / Accepted: 9 February 2009 / Published online: 5 March 2009 © Springer Science + Business Media B.V. 2009

Abstract We determine the degree complexity for all elements of a family k F of birational maps which was introduced and studied in Bedford et al. (Math Phys Anal Geom 11:53–71, 2008). Keywords Birational maps · Degree complexity Mathematics Subject Classification (2000) 37F10

1 Introduction Let P2 denote the complex projective space, and let f : P2 → P2 be a rational map. We will consider its iterates f n = f ◦ f ◦ · · · ◦ f . A basic invariant of iteration is the degree complexity, or the exponential rate of growth: δ( f ) = lim (deg( f n ))1/n . n→∞

(1.1)

Here we consider the family of birational maps k F defined in Section 2 below for an arbitrary polynomial F. If we regard Fa (w) = a0 + a1 w + · · · + a N w N

(1.2)

as depending on the complex parameters a = (a0 , . . . , a N ) ∈ C N+1 , then the dependence a → δ(k Fa ) is lower semi-continuous in the Zariski topology. This means that the set {a : δ(k Fa )  t} is an algebraic variety for all t. In particular, the value of δ(k Fa ) is equal to a constant value δ ∗N outside a proper subvariety of C N+1 .

T. T. Truong (B) Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA e-mail: [email protected]

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A parameter a is said to be exceptional if δ(k Fa ) < δ ∗N . Exceptional maps are of special interest because the lower degree growth indicates the presence of internal symmetries and non-generic behaviors. Such symmetries often make δ more difficult to compute. For instance, there is a birational map K on the projectivized space of q × q matrices (see [8], and [11]). The degree growth of the restriction of the map K to the space of cyclic matrices was shown to be the largest root of the polynomial x2 − (q2 − 4q + 2)x + 1 (see [9]). However, the degree growth of the same map K, restricted to the smaller space of cyclic, symmetric matrices, depends in a much more complicated way on the number q (for primes q it was determined in [4], and for general q it was determined in [6]). In the case of the family k Fa , the numbers δ ∗N were determined in [7]. Here we consider the map a → δ Fa for the full family; we determine the exceptional values as well as the associated rates of degree growth. Theorem 1 Suppose that Fa is as above, and N = deg(Fa ) is even. If a0 = 2/(m + 1) for some integer m  0, then δ(k Fa ) is the largest root of the polynomial x2m+1 (x2 − (N + 1)x − 1) + x2 + N. Otherwise, δ(k Fa ) = δ ∗N is the largest root of x2 − (N + 1)x − 1. The behavior of this family is more complicated when the degree N is odd. For instance, we have Theorem 2 Suppose that Fa is as above, and N = 3. Then we have the following cases: Case 1: a2 = a3 . If a0 = 2/(1 + m) for some integer m  0, then δ(k F ) is the largest root of the polynomial x2m+1 (x3 − 3x2 − 4x − 1) + x3 + x2 + 3x + 2. Otherwise, δ(k F ) = δ3∗ is the largest root of the polynomial x3 − 3x2 − 4x − 1. Case 2: a2 = a3 . 2a. If a0 = 2, then k F is an automorphism, δ(k F ) = 1. Moreover the degree growth is quadratic. 2b. If a0 = 2/(1 + m) for some integer m  1, then δ(k F ) is the largest root of the polynomial x2m (x3 − 3x2 − 2x − 1) + x2 + x + 3. l 2c. If a0 = 2 + 2(1+l) for some integer l  1, then δ(k F ) is the largest root of the polynomial x2l+2 (x3 − 3x2 − 2x − 1) + 3x2 + x + 1. 2d. Otherwise, δ(k F ) is the largest root of the polynomial x3 − 3x2 − 2x − 1. So we see that for case N = 3, there are (infinitely many) linear functions L1 , L2 , L3 , . . . depending on the variable a = (a0 , a1 , a2 , a3 ), and the different cases are determined by conditions of the form Ls (a) = 0 for certain values of s, and Lt (a) = 0 for certain values of t. Thus the sets of exceptional parameters are constructed by linear functions.

Degree Complexity of a Family of Birational Maps

159

We will find in Section 5 that this is typical of the general case for N odd. We also find that there are no automorphisms in the family k Fa other than the ones given in [7]. One difference between the cases when degree N is even or odd is the following. When N is even, the exceptional cases are characterized by a single condition whether a0 = 2/(1 + m) for some integer m  0 or not. When N is odd there is in addition other conditions for exceptional cases, the number of these exceptional conditions are (N + 3)/2. In the proofs of Theorems 1 and 3, following the general frame of Diller and Favre in [12] for working with birational maps of a surface, we will construct spaces Z which is a composition of finite point-blowups of P2 , whose induce map k Z is good (say, A.S. or 1-regular, see [13] for details). We mention here a special phenomena that happens when N is odd: if j exceptional conditions are satisfied, we need to construct spaces Z 1 , . . . , Z j where each Z l+1 is a composition of two pointblowups of Z l . In other words, if N is odd, when a new exceptional condition occurs, we need to blowups two more points.

2 Properties of k F With F as in (1.2), we define two involutions: jF (x, y) = (−x + F(y), y),

  y x−1 , −y − 1 − . i(x, y) = 1 − x − y x−1

and we set k = k F = jF ◦ i. We recall the following sets from [7]: C1 = {x0 = 0},

C2 = {x0 = x1 },

C3 = {x2 = 0},

C4 = {−x20 + x0 x1 + x1 x2 = 0},     x1 x2 C1 = C1 , C2 = 1 + −F =0 , C3 = C3 , x0 x0       x1 x2 x2 x2 1+ =0 . − 1+ −F C4 = x0 x0 x0 x0 The exceptional hypersurfaces of k F are mapped as k F : C4 → [1 : −1 + a0 : 0] ∈ C3 ,

C1 ∪ C2 ∪ C3 → e1 .

The points of indeterminacy of k F are e1 = [0 : 1 : 0], e2 = [0 : 0 : 1], and e01 = [1 : 1 : 0]. The exceptional curves for k−1 F are mapped as (Fig. 1) k−1 F : C1 ∪ C3  → e1 ,

C2 → e2 ,

C4 → e01 .

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Fig. 1 Indeterminacy points and exceptional curves of k F

3 Degree N is Even Let us start by recalling the space X constructed in Section 3 of [7]. We define the complex manifold π X : X → P2 (see Figure 3.1 in [7]) by blowing up points e1 , p1 , . . . , p N−1 in the following order: i) ii) iii) iv)

blowup e1 = [0 : 1 : 0] and let E1 denote the exceptional fiber over e1 , blowup q = E1 ∩ C4 and let Q denote the exceptional fiber over q, blowup p1 = E1 ∩ C1 and let P1 denote the exceptional fiber over e1 , blowup p j = P j−1 ∩ E1 with exceptional fiber P j for 2  j  N − 1.

Here we use the notational convention that if S is a curve at one stage of the construction, then S will denote its strict transforms at subsequent stages (Fig. 2).

Fig. 2 The space X. All exceptional fibers are lying over e1

Degree Complexity of a Family of Birational Maps

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The coordinate projection at P j (1  j  N − 1) is chosen as follows π j : X (s, u) → [s j+1 u : 1 : s ju] ∈ P2 . In this coordinate P j = {s = 0}. For convenience we will use the notations u ∈ P j or [u] P j to indicate the point of P j which has coordinate (0, u) in this coordinate projection. Let k X := π X−1 ◦ k F ◦ π X denote the induced birational map of X. The exceptional curves for k X are C1 , C2 , C4 , P1 , . . . , P N−2 . The curves C1 , C2 , P1 . . . , P N−2 are mapped to the same point 1/a N ∈ P N−1 , while C4 is mapped to the point [1 : −1 + a0 : 0] ∈ C3 . By Lemmas 3.2 and 3.3 in [7], the only way that an exceptional curve can be mapped to a point of indeterminacy is that a0 = 2/(m + 1) for some integer m  0, and in this case we have k2m+1 C4 = [1 : 1 : 0]. X If a0 = 2/(m + 1) we construct the new manifold Z by blowing up the manifold X at the points r0 = k X (C4 ) = [1 : −1 + a0 : 0] ∈ C3 , q1 = k X (r0 ) ∈ Q,

r1 = k X (q1 ) ∈ C3 ,

... qm = k X (rm−1 ) ∈ Q,

rm = k X (qm ) = [1 : 1 : 0] = e01 ∈ C3 .

Call R0 , Q1 , R1 , . . . , Qm , Rm the exceptional fibers of this blowup. Let k Z be the induced birational map of Z (Fig. 3). Lemma 1 If a0 = 2/(m + 1) then the curves C4 , R0 , Q1 , R1 , . . . , Qm , Rm are not exceptional for k Z . Proof It suffices to check that C4 is not exceptional. We choose a local projection for R0 as Z (s, u) → [1 : −1 + a0 + su : s].

Fig. 3 The space Z when a0 = 2/(m + 1). New exceptional fibers are lying on e01 = rm and its pre-images

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In this coordinate chart R0 = {s = 0}. If we rewrite k[x0 : x1 : x2 ] as 

x20 − x0 x1 − x1 x2 + x0 x1  2   2 x0 − x0 x1 − x1 x2 x0 − x0 x1 − x1 x2 : +F x0 (x1 − x0 ) x0 (x1 − x0 )

k[x0 : x1 : x2 ] = 1 : −1 −

then it can be seen that k Z : C4 [x0 : x1 : x2 ] → a1 +

x1 ∈ R0 . x0 

Hence C4 is not exceptional.

The induced map k Z acts as follows k Z : E1  → E1 ,

P N−1 → P N−1 ,

C1 , C2 , P1 , . . . , P N−2 →

1 ∈ P N−1 , aN

Q → C3 → Q, k Z : C4 → R0 → Q1 → R1 → . . . → Qm → Rm → C4 , k−1 Z : C1 , P1 , . . . , P N−1  → −

1 ∈ P N−1 . aN

If S is a curve in Z , we will use the notation S to denote its class in Pic(Z ). Let H ∈ Pic(Z ) denote the class of a generic line. Then H, E1 , P1 , . . . , P N−1 , Q, Q1 , . . . , Qm , R0 , . . . , Rm form an ordered basis for the space Pic(Z ). The curves C1 , C2 , C3 , C4 can be represented in this basis as

C1 = H − E 1 − Q −

N−1 

m 

j=1

j=1

( j + 1)P j −

Q j,

C2 = H − Rm , C3 = H − E 1 − Q −

N−1 

jP j −

j=1

C4 = 2H − E1 − 2Q −

N−1  j=1

m 

Qj −

j=1

jP j − 2

m 

R j,

j=0 m  j=1

Q j − Rm .

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From this, we see that k∗Z : Pic(Z ) → Pic(Z ) is as follows k∗Z (H) = (2N + 1)H − N E1 − (N + 1)Q − (N + 1)

N−1 

jP j

j=1

− (N + 1)

m 

Q j − (N + 1)Rm ,

j=1

k∗Z (E1 ) = E1 , k∗Z (Q) = C3 = H − E1 − Q −

N−1 

jP j −

j=1

m 

Qj −

j=1

m 

R j,

j=0

k∗Z (P j) = 0, 1  j  N − 2, k∗Z (P N−1 ) = C1 + C2 +

N−1 

P j = 2H − E1 − Q −

j=1

k∗Z (R0 ) = C4 = 2H − E1 − 2Q −

N−1 

jP j −

j=1 N−1  j=1

jP j − 2

m 

m 

Q j − Rm ,

j=1

Q j − Rm ,

j=1

k∗Z (R j) = Q j, 1  j  m, k∗Z (Q j) = R j−1 , 1  j  m. Proof of Theorem 1 If a0 = 2/(1 + m) for any integer m  0, δ(k F ) was computed in [7]. It was shown in this case that δ(k F ) is the largest root of x2 − (N + 1)x − 1. Let us suppose now that a0 = 2/(1 + m) for some integer m  0. Then by j Lemma 1, we see that for every exceptional curve , the images k Z (), j  1, n ∗ are disjoint from the determinacy locus. It follows that (k Z ) = (k∗Z )n for all integer n  1. It follows that δ(k F ) is the spectral radius of k∗Z . Thus it is the largest root of the characteristic polynomial of k∗Z , which is P(x) = −x[x2m+1 (x2 − (N + 1)x − 1) + x2 + N]. 

4 Degree N = 3 In this section we will prove Theorem 2. First we consider the more general case where N  3 is an odd number. First we recall the construction of spaces Y and Y1 constructed in [7]. We start from the space X constructed in the

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previous section. Then the line C1 and all blowup fibers P j (1  j  N − 2) are all exceptional for both k X and k−1 X . Moreover C2 is exceptional for k X : k X : C1 , C2 , P1 , . . . , P N−2 → k−1 X : C1 , P1 , . . . , P N−2  →

1 ∈ P N−1 , aN

1 ∈ P N−1 . aN

Hence when N is odd the image of all exceptional curves of k X coincide with a point of indeterminacy ζ0 = a1N ∈ P N−1 . Let πY : Y → P2 be the blowup of X at the point ζ0 ∈ P N−1 , and let P N be the exceptional fiber. At P N we use the coordinate projection (Fig. 4)

π N : (u, s) ∈ Y → s N (su + ζ0 ) : 1 : s N−1 (su + ζ0 ) ∈ P2 . 2 We set Y1 = Y if a0 = m+1 for every integer m  0. Otherwise, as in the previous section, define π1 : Y1 → P2 to be the blowup of Y at the points

r0 = [1 : −1 + a0 : 0] ∈ C3 , q1 = k X (r0 ) ∈ Q, r1 = k X (q1 ) ∈ C3 , ... qm = k X (rm−1 ) ∈ Q, rm = k X (qm ) = [1 : 1 : 0] ∈ C3 , and call R0 , Q1 , R1 , . . . , Qm , Rm the exceptional fibers of this blowup (Fig. 5). Lemma 2 kY1 maps P N ←→ P N−2 by the following formulas P N u →

−a2N u

P N−2 u → −

Fig. 4 The space Y which is the blowup of X at a point on P N−1

1 ∈ P N−2 , − (N − 1)a N + a N−1

1 + a N−1 u ∈ PN . a2N u

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165

Fig. 5 The space Y1 in case a0 = 2/(1 + m)

Proof (Sketch) First, write

k s N (su + ζ0 ) : 1 : s N−1 (su + ζ0 ) = [z0 (s, u) : z1 (s, u) : z2 (s, u)], then if u ∈ P N its image w ∈ P N−2 under kY1 can be computed as w = lim

s→0

z2N−1 z0N−2 z1

.

Now to compute the image of w ∈ P N−2 we do as follows: If u ∈ P N then applying the above argument to the inverse map k−1 will give its image w = −1 g(u) ∈ P N−2 under the map k−1 Y1 . Now the inverse u = g (w) is the image of 

w ∈ P N−2 under the map kY1 . In a similar way, we also have Lemma 3 If we set ζ1 = − ξ1 =

a N−1 , a2N

−(N − 1)a N + a N−1 , a2N

then kY1 : C1 , C2 , P1 , P2 , . . . , P N−3 → ζ1 ∈ P N , k−1 Y1 : C1 , P1 , P2 , . . . , P N−3  → ξ1 ∈ P N . Hence the map k2Y1 : P N → P N is P N u → u +

(N − 1)a N − 2a N−1 = u + ζ1 − ξ1 ∈ P N . a2N

(4.1)

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From (4.1), we see that the orbit of ζ1 (hence also the orbit of all exceptional curves of kY1 ) is k2m Y1 (ζ1 ) = ζ1 + m(ζ1 − ξ1 ) ∈ P N .

(4.2)

Hence the orbit of all exceptional curves of kY1 will contain a point of indeterminacy iff that indeterminacy point is ξ1 , that is iff ζ1 + m(ζ1 − ξ1 ) = ξ1 . The last condition is satisfied iff ξ1 = ζ1 , that is iff the coefficients a N and a N−1 of the polynomial F N (z) satisfy the linear equation −a N−1 = −(N − 1)a N + a N−1 . N Hence if a N−1 = (N−1)a then the map kY1 satisfies the condition (knY1 )∗ = 2 ∗ n N (kY1 ) for all integer n  1, while if a N−1 = N−1)a then the image of all 2 exceptional curves of kY1 is the point of indeterminacy ζ1 = ξ1 .

Proof of Theorem 2 Let Y1 be as above. Since N = 3 we have ζ1 = − ξ1 =

a2 , a23

−2a3 + a2 . a23

Then (4.2) becomes   2a2 − 2a3 a2 a2 k2m . − =− 2 −m Y a23 a3 a23

(4.3) 

In Case 1: a2 = a3 , it follows that the orbit of exceptional curves of kY1 does not contain a point of indeterminacy. Thus (knY1 )∗ = (k∗Y1 )n for all integer n  1, and so δ(k F ) is the spectral radius of k∗Y , which is the largest root of the polynomial given in the statement of Theorem 2. In Case 2: a2 = a3 , we have ζ1 = −ξ1 = − a13 . Hence ζ1 and ξ1 are both the image of exceptional curves C1 , C2 of kY1 , and the image of the exceptional 2 curve C1 of k−1 Y1 . We define a complex manifold πY2 : Y2 → P by blowing up Y1 at the point − a13 ∈ P3 , and call P4 the exceptional fiber of this blowup. We use a local coordinate projection at P4 as follows:      1 1 1 1 π4 : Y2 (s, u) → s3 s2 u − s + : 1 : s2 s2 u − s + ∈ P2 . a3 a3 a3 a3

Degree Complexity of a Family of Birational Maps

The induced map kY2 is as follows:  kY2 : P4 u → 0 : 1 :

 1 ∈ C1 , −a3 + a1 + a23 u

kY2 : C1 [0 : 1 : u] → 

kY 2

a3 − a1 : C2  → a23

167



1 + (a3 − a1 )u ∈ P4 , a3 u → [0 : 0 : 1] = e2 .

P4

Thus the orbit of the exceptional curve C2 encounters an indeterminacy point. Let πY3 : Y3 → P2 be the complex manifold obtained by blowing up Y2 at 1 two points e2 = [0 : 0 : 1] and a3a−a ∈ P4 , and let E2 and P5 be the exceptional 2 3 fibers of this blowup. We use a local coordinate projection at P5 as    1 1 3 3 2 a3 − a1 Y3 (s, u) → s s u + s −s + a3 a3 a23   1 1 2 3 2 a3 − a1 , −s + :1:s s u+s a3 a3 a23 and use a local coordinate projection at E2 as (Fig. 6) E2 (s, u) → [s : su : 1] ∈ P2 . Then the induced map kY3 is as follows: kY3 : P5 u → −a23 u − a3 + 2a1 + a0 − 4 ∈ E2 , kY 3 : E 2 u  →

−u − a3 + 2a1 − a0 + 1 ∈ P5 , a23

k2Y3 : E2 u → u + 2a0 − 5 ∈ E2 ,   a3 − 2a1 + a0 → 2a0 − 4 ∈ E2 . kY 3 : C 2  → − a23 P5

Fig. 6 The space Y3 in case N = 3 and a2 = a3

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Fig. 7 The space Z in case N = 3, a2 = a3 , and a0 = 2

Hence the orbit of the point 2a0 − 4 ∈ E2 is k2l Y3 (2a0 − 4) = 2a0 − 4 + l(2a0 − 5), l  0.

(4.4)

The point 0 ∈ E2 is a point of indeterminacy for kY3 . l for l  0, then from (4.4), the orbit of the exceptional 2a. If a0 = 2 + 2(l+1) curve C2 of kY3 does not contain the point of indeterminacy 0 ∈ E2 . It follows (knY3 )∗ = (k∗Y3 )n for all integer n  1. Then a computation of k∗Y3 on H 1,1 (Y3 ) similar to that of Section 3 completes the proof of Theorem 4 for this case. l for an integer l  0, then from (4.4) it follows that the 2b. If a0 = 2 + 2(l+1) orbit of C2 contains the point of indeterminacy 0 ∈ E2 . We define a complex manifold π Z : Z → P2 by blowing up Y3 at the points   a3 − 2a1 + a0 p6 = − , s0 = kY3 (s0 ) = [2a0 − 4] E2 , a23 P5

s1 = k2Y3 (s0 ), . . . , s2l = k2l+1 Y3 (s0 ) = [0] E2 , and let P6 , S0 , S1 , . . . , S2l the exceptional fibers of this blowup. Then, as in the proof of Lemma 1, it can be shown that the curves C2 , P6 , S0 , . . . , S2l are not exceptional for k Z . It follows (knZ )∗ = (k∗Z )n for all integer n  1. Then a computation of k∗Z on H 1,1 (Z ) similar to that of Section 3 completes the proof of Theorem 2 for this case (Fig. 7).

5 Degree N is Odd In this section we will describe the degree complexities of all elements of the family k F having odd degrees. For fixed N, define for 0  j  N   j  N −l l (−1) a N−l , (5.1) L j(a0 , a1 , . . . , a N ) = (a N− j + a N− j+1 ) − j−l l=0

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 where nj is the binomial coefficient. These linear functions will determine all exceptional parameters of the family k F when deg(F) = N is odd. Theorem 3 Suppose that N = deg(F)  3 is odd. Define h as the largest integer in [0, N − 2] for which L j(a0 , a1 , . . . , an ) = 0 for all 0  j  h. Then exactly one of the following occurs: Case 1: h < N − 2. If a0 = 2/(1 + m) for some integer m  0, then δ(k F ) is the largest real root of the polynomial (1 + x2m+1 )[x3 − Nx2 − (N − h + 1)x − 1] + (N + 1)x2 + (2N − h + 1)x + N − h. Otherwise δ(k F ) is the largest real root of the polynomial x3 − Nx2 − (N + 1 − h)x − 1. Case 2: h = N − 2. l + 2(1+l) 2a. If a0 = 2/(1 + m) for some integer m  0, and a0 = N+1 2 for some integer l  0, then N = 3, a0 = 2, and the map k F is an automorphism with δ(k F ) = 1. Moreover the degree growth is quadratic. In the remaining cases, we assume that N  5. 2b. If a0 = 2/(1 + m) for some integer m  0, then δ(k F ) is the largest real root of the polynomial x2m (x3 − Nx2 − 2x − 1) + x2 + x + N. l 2c. If a0 = N+1 + 2(1+l) for some integer l  0, then δ(k F ) is the 2 largest real root of the polynomial x2l+2 (x3 − Nx2 − 2x − 1) + Nx2 + x + 1. 2d. Otherwise, δ(k F ) is the largest real root of the polynomial x3 − Nx2 − 2x − 1.

The proof of this theorem will be given in Appendix 2, but here we discuss how the linear functions L j are derived. Since F N (z) is a polynomial of degree N, the function 

1 s F −1 − s N



  1 + (1 + s)s F s N

is a polynomial of degree N + 1, and we have     N  1 1 s N F −1 − + (1 + s)s N F = a0 s N+1 + L j(a0 , . . . , a N )s j. s s j=0

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The numbers ζ1 and ξ1 in the previous section can be constructed as follows:    1 1 1+s N − (1 + s)s F = ζ1 − u + O(s), 2 s a N s ζ0 + su    1 1 1+s N + s F −1 − = ξ1 − u + O(s). s a2N s ζ0 + su Then 1





1 s F −1 − ζ1 − ξ1 = − 2 s aN s N



  1 + (1 + s)s F + O(s). s N

Hence ζ1 − ξ1 = −L1 (a0 , . . . , a N )/a2N , so the vanishing of L1 corresponds to the case ζ1 = ξ1 . If ζ1 = ξ1 and N  5, define complex numbers ζ2 and ξ2 as follows    1 1+s 1 N − (1 + s)s F = ζ2 − u + O(s), s a2N s2 ζ0 + sζ1 + s2 u    1 1 1+s N + s = ξ2 − u + O(s). F −1 − s a2N s2 ζ0 + ζ1 s + s2 u Then ζ2 − ξ2 = −L2 (a0 , . . . , a N )/a2N (ζ2 and ξ2 will play the similar roles to that of ζ1 and ξ1 ). However L2 = nL1 /2 = 0 hence ζ2 = ξ2 . Then, if we define ζ3 and ξ3 by    1 1+s 1 N − (1 + s)s F = ζ3 − u + O(s), 2 3 ζ + ζ s + ζ s2 + s3 u s aN s 0 1 2    1 1+s 1 N + s F −1 − = ξ3 − u + O(s). s a2N s3 ζ0 + ζ1 s + ζ2 s2 + s3 u we have ζ3 − ξ3 = −L3 (a0 , . . . , a N )/a2N . Note that now L3 is not a linear combination of L1 and L2 , hence in general ζ3 = ξ3 . Continuing, we assume that ζ1 = ξ1 , ζ2 = ξ2 , ζ3 = ξ3 and N  7. Then we can define ζ4 and ξ4 in the same manner, and ζ4 − ξ4 = −L4 (a0 , . . . , a N )/a2N . Note in this case that L4 is a linear combination of L1 , L2 and L3 . Hence ζ4 = ξ4 . We can continue defining ζ5 and ξ5 , and have that ζ5 − ξ5 = −L5 (a0 , . . . , a N )/a2N . Moreover L5 is not a linear combination of L1 , L2 , L3 and L4 . In Appendix 1 we will show that L2 j is a linear combination of L1 , . . . , L2 j−1 , while L2 j+1 is not a linear combination of L1 , . . . , L2 j for all j. As a consequence of Theorem 3, we show that there are no automorphisms in the family k F other than the ones given in [7]. The following lemma, which is used in the proof of Theorem 4, and also the proof of Theorem 4, are suggested to us by the referee. Lemma 4 a) Let f : Z → Z be an automorphism of a surface Z . Let δ( f ) be the spectral radius of the linear map (which is also the complexity degree of f ) f ∗ :

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H 1,1 (Z ) → H 1,1 (Z ). Then either δ( f ) = 1, or the minimal polynomial p(x) of δ( f ) is symmetric. That is, if d is the degree of p(x) we have xd p(1/x) = p(x). b) Let f : Z → Z be a birational map of a surface Z . Assume that f is 1-regular and that f is birational equivalent to an automorphism. Let χ( f )(x) be the characteristic polynomial of the linear map f ∗ : H 1,1 (Z ) → H 1,1 (Z ). Then roots of χ( f )(x) are δ( f ), 1/δ( f ), and/or 0, and/or algebraic numbers of complex modulus 1. In particular, let g(x) be a factor of χ( f )(x) which is a monic polynomial and whose coefficients are integers and such that g(0) = 0. Then g is either symmetric or anti-symmetric. That is if d is the degree of g(x) we have: either xd g(1/x) = g(x), or xd g(1/x) = −g(x). Moreover if g is anti-symmetric then g(1) = 0. Proof We remark first that if f : Z → Z is a birational map of a surface Z , then χ( f )(x) is a monic polynomial (that is it is a polynomial with integer coefficients, and its leading coefficient is 1). Hence if λ ∈ Q is a root of χ( f )(x) then either λ ∈ N or 0. a) Since f : Z → Z is an automorphism, both f and f −1 are 1-regular. Moreover ( f −1 )∗ = ( f ∗ )−1 and δ( f ) = δ( f −1 ). By ( f −1 )∗ = ( f ∗ )−1 we have that if λ is any root of χ( f ) then 1/λ is a root of χ( f −1 ), and vice versa. In particular, 1/δ( f ) is also a root of χ( f ). We claim that roots of χ( f ) are δ( f ), 1/δ( f ), and/or algebraic numbers of complex norm 1. We have two cases: –Case 1: δ( f ) = 1. Then χ( f ) can not have roots λ with |λ| < 1. Because otherwise then 1/λ is a root of χ( f −1 ) with |1/λ| > 1, which is a contradiction to δ( f −1 ) = δ( f ) = 1. –Case 2: δ( f ) > 1. Then from Theorem 5.1 in [12], δ( f ) is a root of multiplicity 1 and it is the only root λ of χ( f ) with |λ| > 1. The same is true for χ( f −1 ). Hence using the observation above about roots of χ( f ) and χ( f −1 ) we conclude that χ( f ) has no other roots of complex norm not equal to 1 other than δ( f ) and 1/δ( f ). Now we complete the proof of a). Assume that δ( f ) > 1. Then by the remark from the beginning of the proof, δ( f ) ∈ / Q (other wise 1 > λ = 1/δ( f ) is also a rational root of χ( f )(x) which is contradict to that remark). Let p(x) be the minimal polynomial of δ( f ). Then it is a root of χ( f ), and

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all of its coefficients are integers. Now use Case 2 above we now show that p(x) is symmetric. We have

|α| = | p(0)|  1, α: p(α)=0

here one of the α’s is δ( f ), another of the α’s is 1/δ( f ), and the others are algebraic numbers of complex norm 1. This proves a). b) The proof of b) use the proof of a) and is similar to that of a). 

Theorem 4 Suppose N = deg(F)  2. Assume that the map k F is birationally conjugate to an automorphism. Then N = 3, and the map F is that described in Case 2a) of Theorem 3. Proof In Theorems 1 and 3, we constructed spaces Z for which the induced map k Z is 1-regular, and introduced polynomials h(x) such that h(x) = (x − 1)2 g(x) where g(x) factors of the corresponding characteristic polynomials of k Z which has δ(k F ) as a root and g(0) = 0. Hence we can apply Lemma 4 b) to rule out cases for which k F can not be birationally conjugate to an automorphism. -Case 1: N  2 is even. In this case we have two subcases, which for convenience we list in the same order to that of the statement of Theorem 1: Subcase 1a) In this case g(x) = x2m+1 (x2 − (N + 1)x − 1) + x2 + N. Then g(x) is neither symmetric nor anti-symmetric. Hence by Lemma 4, k F does not conjugate with an automorphism. Subcase 1b): In this case g(x) = x2 − (N + 1)x − 1. Although in this case g(x) is anti-symmetric, we see that g(x) is irreducible and has two roots λ and −1/λ. If k F was to be conjugate to an automorphism, by Lemma 4, the two roots of g(x) should be λ and 1/λ. Hence in this case k F does not conjugate to an automorphism. -Case 2: N  3 is odd. We have several subcases, which for convenience we list in the same order that of the statement of Theorem 3: Subcase 2.1 a) In this case g(x) = (1 + x2m+1 )[x3 − Nx2 − (N − h + 1) x − 1] + (N + 1)x2 + (2N − h + 1)x + N − h. Since N − h  2, g(x) is neither symmetrical or antisymmetrical. Hence as in Case 1a), k F does not conjugate to an automorphism. Subcase 2.1 b) In this case g(x) = x3 − Nx2 − (N + 1 − h)x − 1. In this case g(x) can not be symmetric. Although g(x) can be anti-symmetric but we always have g(1) = −N − (N + 1 − h) < 0. Hence from Lemma 4, it follows that k F does not conjugate to an automorphism.

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Subcase 2.2 a) It is proved in [7] that in this case k F does conjugate to an automorphism. For the other subcases 2.2 b, c, d, it can be easily seen that g(x) is neither symmetric nor anti-symmetric. Hence in these cases, k F does not conjugate to an automorphism. 

Acknowledgements The author would like to thank Professor Bedford for his many helpful suggestions. We also would like to thank the referee for her/his helpful comments, in particular for suggesting the proof of Theorem 4 based on Lemma 4 that we include in this paper.

Appendix 1: A System of Linear Equations In this section we explore the system of linear equations defined in in (5.1). Functions L j = L j(a0 , . . . , an ) for some first values of j are: L0 = an + [−an ] = 0, L1 = (an + an−1 ) + [−nan + an−1 ] = −(n − 1)an + 2an−1 ,        n n−1 n−2 n L2 = (an−1 + an−2 ) + −an + an−1 − an−2 = L1 . 2 1 0 2 We will explore the properties of systems of linear equations of the form L j(a0 , a1 , . . . , an ) = 0

(6.1)

for all j = 0, 1, 2, . . . , m, where 0  m < n is a constant integer. It will be convenient to write equations (6.1) as     n n−1 − (an− j + an− j+1 ) = −an + an−1 + ... + j j−1   n− j j+1 (6.2) + (−1) an− j 0 Changing the order of indexes (b j := an− j), the equations (6.2) can be written in a more convenient form       n n−1 n− j j+1 −(b j + b j−1 ) = −b 0 + b1 + . . . + (−1) b j . (6.3) j j−1 0 Lemma 5 If 0  m < n, and m is odd, and if b 0 , b1 , . . . , b n satisfy the equations (6.3) for all j = 1, 3, 5, . . . , m then b 0 , b1 , . . . , b n also satisfy (6.3) for all j = 0, 2, 4, . . . , m + 1. Proof Fixed 0  m < n, where m is odd. Let b 0 , b1 , . . . , b n satisfy the equations (6.3) for all j = 1, 3, 5, . . . , m. To prove Lemma 5 it suffices to prove the following claim: Claim 1: b 0 , b1 , . . . , b n also satisfy (6.3) for j = m + 1.

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The proof is divided in several steps.

i) Reduction 1: In equations (6.3) with j = 1, 3, . . . , m, pushing all b i with i odd to the left hand-sided and pushing all b i with i even to the right hand-sided we can rewrite them as 

 n−1 2b1 = b 0 , 1       n−1 n n−3 b1 + 2b3 = b 0 + b2 , 2 3 1           n−1 n−3 n n−2 n−5 b1 + b3 + 2b5 = b 0 + b2 + b4 , 4 2 5 3 1 .. .



     n−1 n−3 n−m+2 + b3 + . . . + bm−2 + 2bm m−1 m−3 2         n n−2 n−m+3 n−m = b0 + b2 + . . . + bm−3 + bm−1 . m m−2 3 1 b1

The equation (6.3) for j = m + 1 which we want to prove in Claim 1 can be written as 

       n−1 n−3 n−m+2 n−m+1 + b3 + . . . + bm−2 + bm m m−2 3 1       n n−2 n−m+1 = b0 + b2 + . . . + bm−1 . m+1 m−1 2 b1

ii) Reduction 2: For any value of b 0 , b2 , b4 , . . . , bm−1 there exists a unique solution b1 , b3 , . . . , bm to the system (6.3) for j = 1, 3, . . . , m. For a proof of this claim we can use the rewritten system in Reduction 1. iii) Reduction 3: Claim 1 is true in general case if we can show that it is true for the special case b 0 = 1, b2 = b4 , . . . = 0. For a proof use the special structure of the rewritten system in Reduction 1. From now on in this proof we will assume that b 0 = 1, b2 = b4 = . . . = 0. We rewrite Reduction 1 as

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iv) Reduction 4: In equations (6.3) with j = 1, 3, . . . , m, pushing all b i with i odd to the left hand-sided and pushing all b i with i even to the right hand-sided we can rewrite them as   n−1 2b1 = , 1     n−1 n + 2b3 = b1 , 2 3       n n−1 n−3 , + b3 + 2b5 = b1 5 4 2  b1







n−1 n−3 + b3 + . . . + bm−2 m−1 m−3

.. .    n n−m+2 . + 2bm = m 2



The equation (6.3) for j = m + 1 which we want to prove in Claim 1 can be written as     n−1 n−3 b1 + b3 + ... + m m−2       n−m+2 n−m+1 n + bm = . + bm−2 3 1 m+1 v) Reduction 5: Define β1 =

b1 , n

β3 =

b3 , n(n − 1)(n − 2)

β5 =

b5 , n(n − 1)(n − 2)(n − 3)(n − 4) ...

then β1 , β3 , β5 , . . . satisfy the following system of equations 2β1 = 1 − 1 β1 + 2β3 = , 2! 3! β1 β3 1 + + 2β5 = , 4! 2! 5! ..., β3 βm−2 β1 1 + + ... + 2βm = . (m − 1)! (m − 3)! 2! m!

1 , n

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What we want to prove in Claim 1 can be written as   β1 1 β3 βm−2 1 + + ... + + βm 1 + = m! (m − 2)! 3! n−m (m + 1)! vi) Reduction 6: A universal system of linear equations Let θ1 , θ3 , θ5 , . . . be the unique sequence satisfying the following system of infinitely many linear equations 2θ1 = 1, θ1 + 2θ3 = 0, 2! θ3 θ1 + + 2θ5 = 0, 4! 2! ...,

Then, for any sequence c1 , c3 , c5 , . . ., the unique solution to 2z1 = c1 , z1 + 2z3 = c3 , 2! z1 z3 + + 2z5 = c5 , 4! 2! ...,

is z1 = c1 θ1 , z3 = c3 θ1 + c1 θ3 , z5 = c5 θ1 + c3 θ3 + c5 θ1 , ... vii) Reduction 7: Let α1 , α3 , . . . be the unique sequence satisfying the following system 1 , 1! α1 1 + 2α3 = , 2! 3! α1 α3 1 + + 2α5 = , 4! 2! 5! ... 2α1 =

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Then it is easy to see that for β j in Reduction 4: 1 θ j, n for all j = 1, 3, . . . , m, and what we wanted to prove in Claim 1 becomes     θ3 θm−2 θm θm 1 θm 1 θ1 + +. . .+ + − + αm − = 0. − n m! (m − 2)! 3! 1! m n−m m βj = αj −

Hence Claim 1 is proved if we can prove the following claim Claim 2: For any m ∈ N, m odd then the following conclusions are true θ3 θm−2 θm θm θ1 + + ... + + − = 0, m! (m − 2)! 3! 1! m

(6.4)

and αm −

θm = 0. m

(6.5)

viii) Proof of Claim 2: Define a formal series θ(t) = θ1 − t2 θ3 + t4 θ5 − t6 θ7 + . . . From the Reduction 6:   t4 t6 t2 1 = θ(t). 2 − + − . . . = θ(t).(1 + cos t). 2! 4! 6! Hence θ(t) =

1 . 1 + cos t

Similarly, if we define α(t) = tα1 − t3 α3 + t5 α5 . . . then from Reduction 7 α(t) =

sin t . 1 + cos t

It follows that dα = θ(t), dt which proves (6.5). From Reductions 6 and 7 we have θ1 θ3 θm−2 θm αm = + + ... + + . m! (m − 2)! 3! 1! This equality and (6.5) imply (6.4). Hence we completed the proof of Lemma 5.



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Lemma 6 Let n  3 be an odd integer. Let a0 , . . . , an be a solution of the system of linear equations L j(a0 , a1 , . . . , an ) = 0 for all j = 0, 1, 2, . . . , n − 1. Then n  (−1) ja j = 0. j=2

Proof To prove the equality we need only to take the difference between the sum of odd-th equations and the sum of even-th equations. 

Appendix 2: Proof of Theorem 3 Proof The proof is divided into some steps. Step 1: If h < N − 2, we construct a sequence Y1 , Y2 , . . . , Yh+1 where Y j+1 → Y j is a blowup of Y j at a point ζ j ∈ P N−1+ j, where P N−1+ j is the exceptional fiber of the blowup Y j → Y j−1 . Here ζ j’s are constructed inductively in the same way as ζ1 , ζ2 , ζ3 in Section 5. We use the coordinate projection at P N−1+ j as follows (u, s) ∈ Y j → [s N (ζ0 + sζ1 + + . . . s j−1 ζ j−1 + s ju) : 1 : s N−1 (ζ0 + sζ1 + . . . s j−1 ζ j−1 + s ju)] ∈ P2 . The induced map kYh+1 is as follows (see Lemma 2): kYh+1 : C1 , C2 , P1 , . . . , P N−1−(h+1)−1 → ζh+1 ∈ P N−1+h+1 , k−1 Yh+1 : C1 , P1 , . . . , P N−1−(h+1)−1  → ξh+1 ∈ P N−1+h , where ζh+1 and ξh+1 are constructed in the same way as ζ1 , ζ2 , ζ3 . Moreover kYh+1 : P N−1+(h+1) ←→ P N−1−(h+1) is P N−1+(h+1) u →

(−1) N−(h+1) ∈ P N−1−(h+1) , −a2N u + a2N ξh+1

P N−1−(h+1) u →

(−1) N−(h+1) + ζh+1 ∈ P N−1+(h+1) . −a2N u

Step 2: The case when h = N − 2 can be treated as the case when a2 = a3 in Theorem 2. We construct a sequence Y1 , Y2 , . . . , Y N−1 as in Step 1. Then the induced map kY N−1 is as follows kY N−1 : P N−1+N−1 ←→ C1 , kY N−1 : C2 → ζ N−1 → e2 = [0 : 0 : 1],

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where ζ N−1 ∈ P N−1+N−1 is constructed as ζ1 , ζ2 , ζ3 . Hence we see that the orbit of the exceptional curve C2 contains the indeterminacy point e2 . Let Y N → Y N−1 be the blowup of two points ζ N−1 ∈ P N−1+N−1 and e2 , and call P N−1+N and E2 the corresponding exceptional fibers of this blowup. We choose the coordinate projection at P N−1+N as  (u, s) ∈ Y N → s N ζ0 + sζ1 + . . . s N−1 ζ N−1 + s N u : 1 

: s N−1 ζ0 + sζ1 + . . . s N−1 ζ N−1 + s N u ∈ P2 , and the coordinate projection at E2 as (u, s) ∈ Y N → [s : su : 1] ∈ P2 . Using computations as in Lemma 2 we can show that in case h = N − 2 then the induced map kY N : P2N−1 ←→ E2 is kY N : P2N−1 u → −a2N u + a2N ξ N − (N + 1) ∈ E2 , kY N : E 2 u  →

−u + a2N ζ N + 1 ∈ P2N−1 . a2N

Here ζ N and ξ N is constructed in similar manner to that of ζ1 , ξ1 , ζ2 , ξ2 . That the point 0 ∈ E2 is the unique indeterminacy point of kY N lying on E2 is not hard to see. It also easy to see that C2 is an exceptional curve for kY N . We have C2 ∩ E2 = 1 ∈ E2 , which is a regular point of the map kY N . Hence kY N (C2 ) = kY N ([1] E2 ) = ζ N . The map k2Y N : E2 → E2 is u → u + a2N (ξ N − ζ N ) − (N + 2), and kY N (ζ N ) = a2N (ξ N − ζ N ) − (N + 1). When h = N − 2, Lemma 5 implies L j = 0 for all j = 1, . . . , N − 1. From the formulas for ξ N and ζ N , it follows that a2N (ξ N − ζ N ) = 2a0 +

N  (−1) ja j. j=2

Lemma 6 implies a2N (ξ N − ζ N ) = 2a0 . Hence the orbit of C2 is k2l+2 Y N : C2  → 2a0 (l + 1) − (N + 1)(l + 1) − l ∈ E2 . Hence this orbit contains a point of indeterminacy point of kY2 iff that point is 0 ∈ E2 , that is iff there exists an integer l  0 for which 2a0 (l + 1) − (N + 1)(l + 1) − l = 0. The latter condition is exactly the cases 5 and 6 of Theorem 3. If this is the case, then we construct a space Z as the blowup of Y N at the points ζ N ∈ P N−1+N , kY N (ζ N ) ∈ E2 , k2Y N (ζ N ) ∈ P N−1+N , . . . , k2l+1 Y N (ζ N ) = 0 ∈ E2 as in the proof of of Theorem 2. Then the induced map k Z is good, that is it satisfies (k∗Z )n = (knZ )∗ for all integer n  0. Hence the spectral radius of k∗Z is δ(k F ). 

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References 1. Abarenkova, N., Angles dAuriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: From YangBaxter equations to dynamical zeta functions for birational transformations. In: Statistical Physics on the Eve of the 21st Century. Series on Advances in Statistical Mechanics, vol. 14, pp. 436–490. World Sci., River Edge (1999) 2. Abarenkova, N., Angles dAuriac, J.-C., Boukraa, S., Hassani, S., Maillard, J.-M.: Rational dynamical zeta functions for birational transformations. Physica A 264, 264–293 (1999) 3. Angles d’Auriac, J.C., Maillard, J.M., Viallet, C.M.: A classification of four-state spin edge Potts models. J. Phys. A 35, 9251–9272 (2002) 4. Angles d’Auriac, J.C., Maillard, J.M., Viallet, C.M.: On the complexity of some birational transformations. arXiv: math-phys/0503074 5. Bedford, E., Diller, J.: Dynamics of a two parameter family of plane birational maps: maximal entropy. J. Geom. Anal. 16(3), 409–430 (2006) 6. Bedford, E., Kim, K.-H.: Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices. Discrete Contin. Dyn. Syst. arxiv:math.DS/0512507 7. Bedford, E., Kim, K., Truong, T.T., Abarenkova, N., Maillard, J.-M.: Degree complexity of a family of birational maps. Math. Phys. Anal. Geom. 11, 53–71 (2008) 8. Bellon, M.P., Maillard, J-M., Viallet. C.M.: Integrable Coxeter groups. Phys. Lett. A 159, 221–232 (1991) 9. Bellon, M., Viallet, C.M.: Algebraic entropy. Commun. Math. Phys. 204, 425–437 (1999) 10. Boukraa, S., Hassani, S., Maillard, J.-M.: Product of involutions and fixed points. Alg. Rev. Nucl. Sci. 2, 1–16 (1998) 11. Boukraa, S., Maillard, J.-M.: Factorization properties of birational mappings. Physica A 220, 403–470 (1995) 12. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Am. J. Math. 123, 1135–1169 (2001) 13. Fornaess, J.-E., Sibony, N.: Complex dynamics in higher dimension, II. In: Modern Methods in Complex Analysis. Annals of Mathematics Studies, vol. 137, pp. 135–182. Princeton University Press, Princeton (1995)

Math Phys Anal Geom (2009) 12:181–200 DOI 10.1007/s11040-009-9058-y

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy Maarten Bergvelt · Michael Gekhtman · Alex Kasman

Received: 15 June 2008 / Accepted: 18 February 2009 / Published online: 11 March 2009 © Springer Science + Business Media B.V. 2009

Abstract Pairs of n × n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r × r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. Keywords Bispectrality · Integrable systems · Non-commutative KP hierarchy · Spin generalized Calogero-Moser particle system Mathematics Subject Classifications (2000) Primary 37K10 · Secondary 15A24 · 34L99 · 37J35

M. Bergvelt Department of Mathematics, University of Illinois, Urbana, IL 61801, USA e-mail: [email protected] M. Gekhtman Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA e-mail: [email protected] A. Kasman (B) Department of Mathematics, College of Charleston, Charleston, SC 29424, USA e-mail: [email protected]

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1 Introduction 1.1 Background Let CMn = {(X, Z ) | X, Z ∈ Mn×n , rank ([X, Z ] − I) = 1}

(1)

be the set of pairs of complex n × n matrices whose commutator differs from the identity by a matrix of rank one. This space arises naturally in the study of the integrable Calogero-Moser-Sutherland particle system [25, 30]. In particular, the eigenvalues of the time dependent matrix X − itZ i−1 move according to the ith Hamiltonian of this integrable hierarchy and even allows the continuation of the dynamics through collisions [22, 32]. The KP hierarchy is the collection of nonlinear partial differential equations ∂ L = [(Li )+ , L], ∂ti

i = 1, 2, 3. . . .

(2)

for a monic pseudo-differential operator L of order one whose coefficients are scalar functions depending on the time variables ti [27, 28]. If the coefficients of L are further assumed to be rational functions of t1 which vanish as t1 → ∞, then the solutions can be written in terms of the matrices in CMn and the poles move according to the dynamics of the Calogero-Moser-Sutherland system [1, 23, 29, 32]. This was interpreted as a special case of a more general relationship between “rank one conditions” and the KP hierarchy in [14]. Although it seems at first to be quite different in nature, having no obvious dynamical interpretation, the bispectral problem [18] turns out to be another aspect of this relationship between the KP hierarchy and the Calogero-MoserSutherland particle system. As originally formulated in [9], the bispectral problem seeks to find scalar coefficient ordinary differential operators L and  in the variables x and z respectively such that there is a common eigenfunction ψ(x, z) satisfying the eigenvalue equations Lψ = p(z)ψ

ψ = π(x)ψ

(3)

for non-constant functions p and π . As it turns out, if one additionally requires the operator L to commute with another ordinary differential operator of relatively prime order, the solutions to the bispectral problem are exactly the same as the rational solutions to the KP hierarchy mentioned above [31]. (Specifically, up to trivial renormalizations, the bispectral operators are the ordinary differential operators that commute with the pseudo-differential operator L with the identification x = t1 .) Moreover, the bispectral property for these operators is a manifestation of the self-duality of this classical particle system, i.e. the involution on CMn given by (X, Z ) → (Z  , X  ) which linearizes the dynamics of the particle system [19, 32]. In [32] and the conference proceedings [33], Wilson suggests that the correspondence should generalize naturally to the case in which the “rank

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one condition” (1) is replaced with a “rank r condition”. Indeed, various authors have demonstrated that a similar relationship exists between the matrices whose commutator differs from the identity by a matrix of rank r (1 < r  n), the “spin generalization” of the integrable particle system, and matrix generalizations of the KP hierarchy. In particular, the spin generalized system [15] was shown to be related to the matrix KP equation in [24] and to the multi-component KP hierarchy in [2], and rather general rank r conditions were shown to produce solutions to the matrix potential KP hierarchy in [8]. Furthermore, there has been progress on connections between bispectrality for matrix operators and integrable dynamical systems in both classical and quantum settings [7, 11, 12, 34]. None of these, however, has specifically addressed the question of whether and how correspondence between bispectrality and self-duality of the classical Calogero system generalize to the matrix case. 1.2 Outline Section 1.3 will introduce a notation to differentiate between the actions of matrix coefficient differential operators from the left and right sides. The main result of this paper will use this distinction in a generalization of the results on bispectrality to the spin version of the particle system and the matrix KP hierarchy. Section 2 introduces the generalization of (1) to the case of arbitrary rank and relates it to the dynamics of the spin Calogero particle system. Special attention is paid to the block decompositions of the associated operators corresponding to the generalized eigenspaces of the matrix Z . A wave function and pseudo-differential operator are constructed from a choice of n × n matrices satisfying the rank r condition in Section 3. This r × r matrix pseudo-differential operator is shown to satisfy the Lax equation of the KP hierarchy (2). A key component of the proof is the explicit construction of an rn-dimensional space of finitely supported distributions in the spectral parameter which annihilate the wave function. Several obvious group actions on the space of matrices satisfying the rank r condition are investigated in Section 4 with emphasis on their effect on the corresponding KP solution. Of special interest is the bispectral involution which has the effect of exchanging variables and transposing the wave function. The main theorem is the construction in Section 5 of commutative rings of matrix differential operators in x and z and the demonstration that they have the KP wave function as a common eigenfunction. A final section contains closing remarks and lists problems for future research on this topic. 1.3 Notation We will make use of the notation ML and MR to distinguish between the cases in which the operator M is acting from the left or the right, respectively. So,

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for instance, if M and P are both r × r matrices, then [M, P] = (ML − MR )P. Similarly, if L=

N 

Mi (x)∂xi

(4)

i=0

is an ordinary differential operator in x of degree N with coefficients Mi (x) that are r × r matrices and ψ(x) is an r × r matrix function we define LL (ψ) = L(ψ) =

N  i=0



 ∂i Mi (x) ψ(x) , ∂ xi

as usual. However, the operator can also act from the right  N  i  ∂ ψ(x) Mi (x). LR (ψ) = ∂ xi i=0 Equivalently, if we denote by L the differential operator with coefficients Mi that are the ordinary matrix transpose of the coefficients of L, we can say    LR (ψ) = L ψ  . (5) In the rest of the paper we will use the notation Ik for the k × k identity matrix. Also we will abuse notation by using the symbol I to denote the identity transformation on many different vector spaces whenever its use should make it clear which is intended.

2 Spin Calogero Matrices Let sCMrn be the the set of 4-tuples of matrices (X, Z , A, B) such that the n × n commutator [X, Z ] differs from the identity by the rank r matrix B A: sCMrn = {(X, Z , A, B) | X, Z ∈ Mn×n , A, B ∈ Mr×n , [X, Z ]− I = B A = 0}. (6) This space arises naturally in the description of generic initial conditions of the Spin Calogero particles, as we will see below. More importantly, the dynamics linearizes there (when one considers the phase space to be sCMrn modulo the action of GL(n) to be described in the section on symmetries). Let qi (1  i  n) be the distinct positions of n particles on the complex plane, q˙ i be their momenta, and fij = β jαi be their “spins” represented as the products of n column r-vectors αi and n row r-vectors β j subject to the constraint fii = −1.

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We associate to this data the matrices (X, Z , A, B) ∈ sCMrn in the form Z ij = q˙ i δij + (1 − δij)

Xij = qi δij

fij qi − q j

⎞ β1 ⎜ ⎟ B = ⎝ ... ⎠ . ⎛



A = α1 · · · αn



βn

The dynamics of the eigenvalues of X − itZ i−1 are governed by the Hamiltonian H = trZ i . This is the spin Calogero system [15, 24]. In the special case r = 1, this reduces to the more famous (spinless) Calogero-Moser-Sutherland particle system [30]. 2.1 Block Decomposition For a fixed choice of (X, Z , A, B) ∈ sCMrn we get a decomposition of V = Cn into generalized eigenspaces of Z :

V= Vλ Vλ = {v ∈ Cn | (Z − λI)k v = 0 for some k  0}. λ

The restriction of Z to Vλ will be denoted by Z λ = λI + Nλ where I is the identity operator on Vλ and Nλ is nilpotent. Rational expressions in zI − Z λ below will always be interpreted by expansion in positive powers of the nilpotent Nλ . We will utilize subscripts λ and μ which will run over the eigenvalues of Z to similarly denote the blocks of other linear operators associated to this decomposition of Cn . Specifically, Aλ : Vλ → Cr will be the restriction of the map A, Bλ : Cr → Vλ will be the map B followed by projection onto Vλ , and for a linear operator M from Cn to itself (such as X) Mλμ will be the block corresponding to the map from Vμ to Vλ . The sCM condition (6) involves the commutator [X, Z ] = (Z R − Z L )(X). Interestingly, although the operator Z R − Z L = −ad(Z ) is not invertible, its “off-diagonal” action is invertible which allows us to solve for Xλμ when μ = λ. Lemma 2.1 Let λ = μ be generalized eigenvalues of Z . Then  ((Nλ )R − (Nμ ) )k  −1 L Xμλ = (Z λ )R − (Z μ )L B μ Aλ = B μ Aλ . (λ − μ)k+1 k0

Proof Since (Z R − Z L )μλ = (λ − μ)Iμλ + (Nλ )R − (Nμ )L

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differs from the a nonzero multiple of the identity by a nilpotent matrix, it is invertible. Specifically, we may invert it in general using (Z R − Z L )−1 μλ =

 ((Nλ )R − (Nμ ) )k L . (λ − μ)k+1 k0

Applying this when solving [X, Z ] − I = B A for any off-diagonal block of X yields the claimed formula.  It will later be necessary to evaluate residues of matrix functions written in terms of the blocks Z λ . For this purpose the following “obvious” lemma will be useful. For convenience we introduce notation for “divided derivatives”: f [k] =

1 dk f . k! dzk

Lemma 2.2 Let f (z) be a rational function that is regular at z = λ, then 

f (z) Res z=λ (zI − Z λ )k+1



= f [k] (Z λ ).

Proof 

f (z) Res z=λ (zI − Z λ )k+1



 = Res z=λ

 = Res z=λ

(−∂z )k 1 f (z) k! zI − Z λ 1 f [k] (z) zI − Z λ

⎛ = Res ⎝ f [k] (z) z=λ

⎛⎡ = Res ⎝⎣ z=λ

z=λ

f

[k]

z=λ

⎞ (−∂z )s /s!

s 0



⎛

= Res ⎝ f [k] (z)

⎞

∂zs /s! f [k] (z)Nλs ⎦

1 ⎠ z−λ

1 (z + Nλ ) z−λ

 s 0

⎞ Nλs ⎠ (z − λ)s+1

Nλs ⎠ (z − λ) ⎤

s 0

 = Res









= f [k] (Z λ ). 

Of course, in the above lemma f [k] (Z λ ) is a matrix and may not commute with other matrices appearing. So, one needs a little care in applying the Lemma 2.2.

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3 Matrix KP Hierarchy Let ν = (X, Z , A, B) ∈ sCMrn and associate to it the wave function ψν depending on the spectral parameter z and the times t = (t1 , t2 , t3 , . . .):   ˜ −1 (zIn − Z )−1 B , ψν ( t, z) = γ ( t, z) Ir + A X (7) where1 ˜ = X( ˜ t) = X

∞ 

iti Z

i−1

− X,

and

γ ( t, z) = exp

i=1

∞ 

 i

ti z

.

(8)

i=1

(Equation 7 specializes to the familiar determinantal formula from [31] in the case r = 1 as already noted in [33] and [17].) If qν (z) = det(zI − Z ) is the characteristic polynomial of Z then the wave function (7) can be multiplied by qν and an exponential so as to yield a polynomial in z with coefficients that are rational in the times K( t, z) = γ −1 ( t, z)ψν ( t, z)qν (z).

(9)

−1

Indeed, qν (z)(zI − Z ) is the classical adjoint of zIn − Z and hence polynomial in z. Letting ∂ = ∂∂x be the differential operator in x = t1 , we note that the “Sato-Wilson Operator” Kν = K( t, ∂) is an ordinary differential operator satisfying ψν ( t, z) =

1 Kν γ ( t, z). qν (z)

(10)

The main goal of this section is to prove that the pseudo-differential operator Lν = Kν ◦ ∂ ◦ Kν−1

is a solution to the matrix KP hierarchy in that it satisfies the Lax equation (2). As in [31] (see also [19, 28]), the proof will involve identifying finitely supported distributions in z that annihilate the function ψν . 3.1 Conditions Satisfied by ψν In the previously studied rank one case, identifying linear combinations of residues of the wave-function and its derivatives which are uniformly equal to zero had the geometric significance of identifying a point in an infinite dimensional Grassmannian manifold or a section of a trivialized line bundle over a singular curve [19, 23, 31]. This section generalizes this construction to the higher rank case in a way that will prove useful for deriving the Lax equation.

1 Note that the dependence on t in − X ˜ is such that its eigenvalue dynamics are governed by the i ith spin Calogero Hamiltonian.

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Consider a generalized eigenvalue λ of Z with multiplicity and use the notation of Section 2.1 to denote by Z λ , Xλμ , Aλ , etc. the blocks of the operators X, Z , A and B. Let v ∈ Cr + have the decomposition ⎛ ⎞ v0 ⎜ v1 ⎟ ⎜ ⎟ ⎜ v2 ⎟ ⎜ ⎟ v=⎜ . ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎝v −1 ⎠ w where vi ∈ Cr and w ∈ C and define the distribution cv,λ taking r × r matrix functions of z to r component constant vectors by the formula    −1  . (11) (z − λ)i vi cv,λ ( f (z)) = Res f (z) · Aλ (zI − Z λ )−1 w + z=λ

i=0

In this section we will show that there are r linearly independent distributions of this form satisfying cv,λ (ψν ( t, z)) ≡ 0. Consequently, by running through all of the eigenvalues of Z we obtain in this manner an rn-dimensional space of conditions satisfied by the wave function. Indeed, if {λi } are the generalized  eigenvalues of the n × n matrix Z with multiplicities { i }, then n = i . Consider the × (r + ) matrix  

λ = Bλ Nλ Bλ Nλ2 Bλ · · · Nλ −1 Bλ −Xλλ .   Lemma 3.1 If v ∈ ker λ then cv,λ ψν ( t, z) = 0 for all values of the variables t. Proof Note first that ψν ( t, z) has the block decomposition    −1 −1 ˜ )κμ (zI − Z μ ) Bμ Aκ ( X ψν ( t, z) = γ (z, t) I +

(12)

κ,μ

where again the sum is taken over all (not necessarily distinct) pairs of generalized eigenvalues κ and μ of Z . Now, we wish to use Lemma 2.2 to expand the residue in (11) where f (z) is replaced by (12). It will be convenient to introduce the abbreviation  ˜ −1 )κμ , Aκ ( X Cμ = κ

so that we have

 ψν ( t, z) = γ ( t, z) I +



 −1

Cμ (zI − Z μ )

Bμ

(13)

μ

and Aλ =

 μ

˜ μλ . Cμ X

(14)

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The various contributions to the residue are usefully organized according to dependence on Cμ . First of all, there is the contribution independent of Cμ . It is given by   Res γ ( t, z)Aλ (zI − Z λ )−1 w = Aλ γ ( t, Z λ )w. (A) z=λ

Making use of Lemma 2.1 one finds that the contributions containing Cμ for μ = λ are     ˜ μλ γ ( t, Z λ )w. Cμ Res (zI − Z μ )−1 Bμ Aλ (zI − Z λ )−1 wγ ( t, z) = − Cμ X μ=λ

z=λ

μ=λ

(B) ˜ μλ = −Xμλ , as the off-diagonal part of Z vanishes, where for μ = λ we have X see (8). Next we turn to the terms involving Cλ . The first one is   −1  −1 i (z − λ) vi Cλ Res γ ( t, z)(zI − Z λ ) Bλ z=λ

i=0

= Cλ γ ( t, Z λ )

−1 

Nλi Bλ vi .

(C)

i=0

The other term linear in Cλ is   Cλ Res γ ( t, z)(zI − Z λ )−1 Bλ Aλ (zI − Z λ )−1 w z=λ

  = Cλ Res γ ( t, z)(zI − Z λ )−1 ([Xλλ , Z λ ] − I)(zI − Z λ )−1 w z=λ

  = − Cλ Res γ ( t, z)(zI − Z λ )−2 w z=λ

  + Cλ Res γ ( t, z)(zI − Z λ )−1 [Xλλ , Z λ − zI](zI − Z λ )−1 w z=λ

= − Cλ γ ( t, Z λ )w − Cλ Res(γ ( t, z)(zI − Z λ )−1 Xλλ w) z=λ

+ Cλ Res(Xλλ (zI − Z λ )−1 γ ( t, z)w) z=λ

= − Cλ γ ( t, Z λ )w − Cλ γ ( t, Z λ )Xλλ w + Cλ Xλλ γ ( t, Z λ )w ˜ λλ γ ( t, Z λ )w − Cλ γ ( t, Z λ )Xλλ w, = − Cλ X

(D)

Here we use that it follows from (8) that ˜ t)γ ( t, Z ) = γ ( t, Z ) − Xγ ( t, Z ), X( where γ ( t, Z ) =

dγ ( t,z) |z=Z , dz

so the the λλ component of identity (15) is

˜ λλ ( t)γ ( t, Z λ ) = γ ( t, Z λ ) − Xλλ γ ( t, Z λ ). X

(15)

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Since v ∈ ker λ is equivalent to the statement −1 

Nλk Bλ vk − Xλλ w = 0,

(16)

k=0

we see that (C) cancels against the second term in (D). So, combining all four terms gives (A) + (B) + (C) + (D) = Aλ γ ( t, Z λ )w −



˜ μλ γ ( t, Z λ )w − Cλ X ˜ λλ γ ( t, Z λ )w. Cμ X

μ=λ

Applying (14) shows that this is equal to zero as required.



Lemma 3.2 The distributions cv,λ for v ∈ ker λ form an r -dimensional space. Proof Note that the map  : v ∈ ker λ → cv,λ is itself a linear map. What we need to prove, therefore is that dim ker λ − dim ker  = r . A vector v clearly does not lie in the kernel of  if vi = 0 for any i. The dimension of the kernel of  is therefore equal to the dimension of the space of vectors w with the property that Xλλ w = 0 and Aλ (zI − Z λ )−1 w = 0. In fact, we will show that the only such w is the zero vector (and hence that dim ker  = 0). Beginning with the fact that [(zI − Z λ ), Xλλ ] − Bλ Aλ = I. Multiplying by (zI − Z λ )−1 on the right, applying both sides of the resulting equation to w and then multiplying by (zI − Z λ )−1 on the left gives us that Xλλ (zI − Z λ )−1 w = (zI − Z λ )−2 w. Expanding both sides of this equation in terms of powers of (z − λ) and equating like powers gives us that Xλλ Nλk w = kNλk−1 w,

for k > 0.

Since Nλ is nilpotent, for a sufficiently large k the left-hand side is equal to zero. But the equation then tells us that Nλk−1 w is then also equal to zero, which again means that the left hand side would be zero for a smaller value of k. Repeating this process until k = 1 we find that w = 0. A similar argument shows that dim ker λ = r . Considering instead the vectors w such that w λ = 0 implies that w  X = w (zI − Z λ )−1 Bλ = 0 and the same process reveals that w = 0 so that λ has rank . Consequently, its kernel has dimension (r + ) − = r . 

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3.2 The Kernel of Kν The results of the previous section on the distributions annihilating ψν give us information about the kernel of the matrix ordinary differential operator Kν defined in (10): Corollary 3.3 Let cv,λ be as in Lemma 3.1. Then the r component vector valued function φv,λ ( t) = cv,λ



γ ( t, z) qν (z)

 (17)

is in the kernel of the operator Kν . Proof Since cv,λ commutes with multiplication and differentiation in x = t1 , we have   Kν φv,λ = cv,λ Kν γ ( t, z)q−1 ν (z) = cv,λ (ψν ) = 0, 

by (7) and Lemma 3.1.

In fact, the entire kernel of Kν is spanned by functions of this form, and as a consequence they satisfy certain useful linear differential equations. Corollary 3.4 If φ( t) is a vector in the kernel of Kν then it is a linear combination of the φv,λ ( t), and so satisfies the equation ∂ ∂k φ( t) = φ( t). ∂tk ∂t1k Proof By making use of all of the eigenvalues of Z , Corollary 3.3 gives us rn linearly independent vector functions in the kernel of the nth order ordinary differential operator with r × r matrix coefficients. Since they are linearly independent (those corresponding to the same eigenvalue are linearly independent by Lemma 3.2 and those corresponding to different eigenvalues cannot be linearly dependent due to the factor of exλ ) this accounts for the entire kernel of Kν . Note that γ ( t, z) trivially satisfies ∂ ∂k γ ( t, z) = zk γ ( t, z) = γ ( t, z). k ∂tk ∂t1 Now, the proof here is elementary because differentiation in ti commutes with the residue, multiplication by functions of z and matrix multiplication in the definition of φv,λ and applies by linearity to the entire kernel. 

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3.3 The Lax Equation Now we come to main point of this section. If the ν moves according the spin Calogero dynamics the wave function ψν depends on the time variables t, and this produces a solution of the matrix KP hierarchy. More precisely: Theorem 3.5 The pseudo-differential operator Lν = Kν ◦ ∂ ◦ Kν−1 satisfies ∂ Lν = [(Liν )+ , Lν ]. ∂ti Proof First, we note that the (pseudo)-differential operator (Liν )− ◦ Kν is actually a differential operator since (Liν )− ◦ Kν + (Liν )+ ◦ Kν = Kν ◦ ∂ i and therefore (Liν )− ◦ Kν = −(Liν )+ ◦ Kν + Kν ◦ ∂ i . Now, let φ(x) be a vector function in the kernel of the operator Kν . Then applying ∂t∂ i to the equality Kν φ = 0 and using Corollary 3.4 we find 0=

∂ ◦ Kν (φ) = (Kν )ti (φ) + Kν (φti ) ∂ti

= (Kν )ti (φ) + Kν (∂ i φ) = (Kν )ti φ + Liν ◦ Kν (φ) = (Kν )ti (φ) + (Liν )+ ◦ Kν (φ) + (Liν )− ◦ Kν (φ) However, since φ is in the kernel of Kν we know that (Liν )+ ◦ Kν (φ) = 0. (The same cannot be said for the last term as the left factor is a pseudodifferential operator and so the kernel of the differential operator (Liν )− ◦ Kν may not contain the kernel of Kν .) Then the last displayed equality gives us that the entire kernel of Kν is in the kernel of the ordinary differential operator (Kν )ti + (Liν )− ◦ Kν . This operator has order strictly less than n since Kν is monic (so that its derivative with respect to ti has order less than n) and since multiplying by (Liν )− will lower the order. An operator of this order can only have such a large kernel if it is the zero operator and we conclude (Kν )ti = −(Li )− ◦ Kν . Using this we find that (Lν )ti = (Kν )ti ◦ ∂ ◦ Kν−1 − Kν ◦ ∂ ◦ Kν−1 ◦ (Kν )ti ◦ K−1 = −(Liν )− ◦ Kν ◦ ∂ ◦ Kν−1 + Kν ◦ ∂ ◦ Kν−1 ◦ (Liν )− ◦ Kν ◦ Kν−1 = [Lν , (Liν )− ] = [(Liν )+ , Lν ]. 

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4 Symmetries The symmetry X → X + cZ j of sCMrn induces the integrable dynamics of both the particle system and the wave equations of the matrix KP hierarchy. Here are some other symmetries and how they affect the KP Lax operator. 4.1 Action of GL(n) For any G ∈ GL(n) we define SG acting on sCMrn by SG : sCMrn → sCMrn (X, Z , A, B) → (GXG−1 , GZ G−1 , AG−1 , GB). The Lax operator Lν is unaffected by this action: L SG (ν) = Lν . Since this symmetry does not affect the corresponding dynamical objects from the previous sections, it makes sense to consider sCMrn modulo this group action as the phase space of the spin Calogero particle dynamics as well as the corresponding matrix KP solutions. 4.2 Action of GL(r) Clearly, if we have G ∈ GL(r) then we can conjugate solutions to the matrix KP hierarchy to get solutions that are technically different, but not very different. This also manifests itself as a group action on the level of sCMrn . Let G ∈ GL(r) then if sG : sCMrn → sCMrn (X, Z , A, B) → (X, Z , GA, BG−1 ) one finds that LsG (ν) = GLν G−1 . 4.3 Changing r There is an easy way to take an r × r solution and turn it into an R × R solution for r < R. Let a be an R × r and b an r × R matrix such that b a = I is the r × r identity matrix. Defining U a,b : sCMrn → sCMnR by U a,b (X, Z , A, B) = (X, Z , a A, Bb ). one then has LUa,b (ν) ( t) = aLν ( t)b .

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4.4 The Bispectral Involution Finally, we have an important discrete symmetry whose effect on the KP solution will be the subject of the next section:  : sCMrn → sCMrn ν = (X, Z , A, B) → ν  = (Z  , X  , B , A ) The significance of this symmetry on the KP solution is most easily seen by looking at the wave function ψν as a function of x = t1 and z only (setting all of the other times equal to zero). As in the case r = 1 (cf. [32]), it involves an exchange of x and z, but when r > 1 one must also take the transpose of the function: ψν (x, z) = ψν (z, x).

(18)

5 Bispectrality Definition 5.1 A bispectral triple (L, , ψ) consists of a differential operator L in x as in (4), a differential operator  in the variable z also having r × r matrix coefficients, and an r × r matrix function ψ(x, z) of x and z satisfying the equations LL (ψ) = p(z)ψ

and

R (ψ) = π(x)ψ,

(19)

where p(z) and π(x) are non-constant, scalar eigenvalues. This seems to be a natural matrix generalization of the scalar bispectral problem for differential operators considered in [9]. However, we note that this differs from the matrix generalization previously considered by Zubelli [34] in which both differential operators acted from the left, and hence is in some ways more closely related to the work of Grünbaum and Iliev [16] in which the operators (one differential and one a difference operator) acted on a common eigenfunction from opposite sides. We also wish to differentiate (19) from the “bundle bispectrality” considered by Sakhnovich-Zubelli [26] where the operators L and  were allowed to depend on both variables. (Here we are interested only in the case that L is independent of z and  is independent of x.) Let ν ∈ sCMrn and qν (z) = det(zI − Z ). In this section, since the dynamics are not significant, we will consider x = t1 and ti = 0 for i > 1. Thus, for instance, we will write ψν (x, z) for ψν ((x, 0, 0, . . .), z) and γ (x, z) = exz . In the next section we will associate two commutative rings of ordinary differential operators (one acting in x and one acting in z) to the choice of ν ∈ sCMrn and then in the following section we will demonstrate that ψν (x, z) is a common eigenfunction for the operators in the rings. In particular, any operator of order greater than zero from each of the rings along with ψν form a bispectral triple as in Definition 5.1.

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5.1 Commutative Rings of Matrix Differential Operators Associate to ν ∈ sCMrn the ring Rν ⊂ C[z] defined by the property that composition with the polynomials preserves the conditions annihilating ψν ( t, z) from Lemma 3.1:   Rν = p ∈ C[z] | cv,λ (ψν ) = 0 ⇒ cv,λ ( p(z)ψν ) = 0 . Lemma 5.1 The ring Rν is non-empty. In particular, q2ν (z)C[z] ⊂ Rν . Proof Note that qν (z)ψν ( t, z) = Kν γ ( t, z) is non-singular in z. Then the claim follows from the fact that cv,λ (qν (z) f (z)) = 0 for any non-singular function f .  By substituting the pseudo-differential operator Lν into these polynomials, we associate a commutative ring of pseudo-differential operators Rν = { p(Lν )| p ∈ Rν }

to ν. However, as the next lemma demonstrates, these are in fact differential operators. Lemma 5.2 If p ∈ Rν then L = p(Lν ) is a differential operator (as opposed to a general pseudo-differential operator) satisfying the eigenvalue equation Lψν (x, z) = p(z)ψν (x, z). Proof Since the leading coefficient of Kν is a nonsingular matrix (in fact, it is the identity matrix because of the form of ψν ), it is sufficient to show that the kernel of Kν is contained in the kernel of Kν ◦ p(∂) because then we know that this ordinary differential operator factors as L ◦ Kν for some ordinary differential operator L which meets all of the other criteria. So, now let φ(x) be a function in the kernel of Kν . By Lemma 3.2 we know that φ(x) is a linear combination of functions of the form (17). However, Kν ◦ p(∂)cv,λ (γ (x, z)q−1 ν (z)) = cv,λ ( p(z)ψν (x, z)) = 0

by (10) and the definition of Rν .



We will also associate a commutative ring of ordinary differential operators in z to ν. Applying the procedure above to the point ν  = (Z  , X  , B , A ) ∈ sCMrn we have another commutative ring Rν  of differential operators in x. We convert them to differential operators in z by simply replacing x with z, ∂x with ∂z and transposing the coefficients:   Rν = L (z, ∂z )|L(x, ∂x ) ∈ Rν  .

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5.2 Common Eigenfunction Our main result is the observation that ψν (x, z) is a common eigenfunction for the differential operators in the rings Rν and Rν satisfying eigenvalue equations of the form (19): Theorem 5.3 Let p ∈ Rν and π ∈ Rν  , then there exist ordinary differential operators L(x, ∂x ) ∈ Rν and (z, ∂z ) ∈ Rν such that Lψν (x, z) = p(z)ψν (x, z)

and

R ψν (x, z) = π(x)ψν (x, z).

Proof The first equation follows from Lemma 5.2. Similarly, it follows from Lemma 5.2 that there is a differential operator Q(x, ∂x ) with the property that Qψν  (x, z) = π(z)ψν  (x, z). Exchanging the roles of x and z in this equation, taking the transpose, and applying (5) and (18) results in the second equation of the claim. 

6 Example For the sake of clarity, we briefly illustrate the main ideas with an example.2 Consider ν = (X, Z , A, B) ∈ sCM23 where ⎛ ⎞ ⎛ ⎞ 0 0 0 0 1 0 X = ⎝−1 0 −1⎠ Z = ⎝0 0 0 ⎠ 1 0 2 0 0 0 ⎛



 0 1 0 A= 0 0 1 Then

and

 ψν (x, z) = e

xz

 I+

⎞ 0 1 B = ⎝−2 0 ⎠ . 1 −1

−2zx2 +3zx+2x−2 1 (x−2)x2 z2 (x−2)x2 z xz−2 1−x (x−2)xz (x−2)xz2

 .

This can be written as ψν = Kν exz /qν (z) where qν (z) = det(zI − Z ) = z3 and   2(x−1)   3−2x   1 0 1 0 3 (x−2)x2 (x−2)x (x−2)x2 2 ∂ Kν = ∂ + + ∂ 1−x 1 2 0 1 − (x−2)x 0 x−2 (x−2)x is an ordinary differential operator.

2 This

example was essentially chosen at random so as to be representative of the general construction without being either too trivial or too cumbersome to print.

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To find the conditions satisfied by ψν (x, z), we note that the kernel of ⎛ ⎞ 0 1 −2 0 0 0 0 0 0

0 = ⎝−2 0 0 0 0 0 1 0 1 ⎠ 1 −1 0 0 0 0 −1 0 −2 is made up of vectors of the form   v = c1 c2 c22 c4 c5 c6 3c1 + c2 c8 −c1 − c2 . Hence, we conclude that cv,0 (ψν (x, z)) = 0 where    c5 z2 + 12 c2 z + c1 + cz8 cv,0 ( f (z)) = Res f (z) . c6 z2 + c4 z + c2 + −c1z−c2 z=0 Moreover, cv,0 ( p(z)ψν ) = 0 whenever p ∈ z4 C[z] = Rν . Consequently, we can find an ordinary differential operator having any of these polynomials as its eigenvalue. In particular, solving Kν ◦ ∂ 4 = L ◦ Kν

for L we find

 8 3x4 −18x3 +50x2 −63x+30) 4(23x3 −93x2 +138x−72) − ( 4 4 4 5 (x−2) x (x−2) x L= 8(2x3 −5x2 +9x−6) 4 (2x4 −8x3 +27x2 −42x+24) − 4 3 (x−2) x (x−2)4 x4  4(6x3 −27x2 +49x−30)  4(13x2 −35x+26) − 3 3 3 4 (x−2) x (x−2) x + ∂ 4 4x2 −7x+6) 4 (2x3 −6x2 +13x−10) − ( 

(x−2)3 x2

+

 8(x2 −3x+3) − (x−2)2 x2 4 (x−2)2

(x−2)3 x3

4(3 x−4) (x−2)2 x3 4 x2 −2x+2 − ((x−2)2 x2 )



∂2 + ∂4

which satisfies Lψν = z4 ψν . Of course, we can follow this same procedure beginning with another element of sCM23 . In particular, if we begin with ν  = (Z  , X  , B , A ) instead then the differential operator we produce will be     4 16x2 +65x+90) 4(16x2 +65x+90) 80 x+216 − ( − 8(10 xx+27) 6 6 5 5 x x x Q= + ∂ 4 4x2 +21x+36) 4(4x2 +21 x+36) 8x+12 − ( − 4(2x+3) x4



2 12x +65x+90) − ( x4

+

6 x2

 +

x6

2

6 4 − 12 x2 x2 0 4 − x62



40 x+108 x4 2 − 8x +42x+72 x4

x3



∂2 +

 34x+60 x3

0

−

 4(2x+9)

x3 14x+24 x3

x5

∂3

∂ 4 − 4∂ 5 + ∂ 6 .

The function ψν  is an eigenfunction for this operator satisfying Qψν  (x, z) = (4z4 − 4z5 + z6 )ψν  (x, z).

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More interestingly, since ψν  (x, z) = ψν (z, x), if we transpose the matrix coefficients on Q, replace x with z and ∂ = ∂x with ∂z , we get a differential operator  in the variable z. This operator applied to ψν (x, z) (the wave function computed earlier) from the right satisfies R ψν (x, z) = (4x4 − 4x5 + x6 )ψν (x, z), demonstrating bispectrality.

7 Conclusions and Comments The main results of the present paper can be viewed as another step in addressing the “bispectral problem” of F.A. Grünbaum [9, 18], seeking operators satisfying eigenvalue equations of the form (3). In [10], the authors considered the case in which one of the operators is a second order difference operator with matrix coefficients and the two operators act on matrix eigenfunctions from different directions (see also [6]). However, bispectrality for matrix differential operators has only been studied with both operators acting from the left [34].3 Here we consider the case (19) in which the operators are r × r matrix differential operators acting from different directions. Since our construction conveniently reproduces the results of Wilson’s seminal paper [31] in the special case r = 1, this particular formulation of the bispectral problem appears to be the correct one for generalizing those results to the case of matrix differential operators. However, the method of proof and especially the explicit formulation of the “conditions” satisfied by the wave function above are novel even for r = 1. In [31], it was shown that the bispectral operators associated to sCM1n are in fact the only bispectral scalar ordinary differential operators which commute with operators of relatively prime order up to obvious renormalizations and changes of variable.4 By Lemma 5.1 it follows that the differential operators produced by the construction in this paper also all have the property that they commute with other differential operators of relatively prime order. In addition, this paper can be seen as contributing to the literature establishing a link between bispectrality and duality in classical and quantum integrable systems. (See, for instance, [13, 17, 19–21, 32].) Again, the main results of the present paper for the spin Calogero system in the case r = 1 reproduce results previously presented for the spinless case in [19, 32]. Some questions arise naturally which we have not pursued. There are additional commuting Hamiltonians for the spin Calogero system [15] and

3 After

the present paper was submitted for publication, the paper [5] appeared which contains related results on the bispectrality of matrix differential operators acting from different directions. In particular, they obtain the same class of matrix bispectral operators using different methods and in a non-dynamical context. 4 These were called “rank one” operators in that context, but we will avoid that terminology here to avoid confusion with the rank r which is something different.

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corresponding isospectral deformations for the multi-component KP hierarchy [4], but their relationship to bispectrality has not been explored here. We have not looked at the algebro-geometric implications of the rings Rν . Certainly as in the case r = 1 [31, 32], these contain operators of relatively prime order are isomorphic to the coordinate rings of rational curves with only cuspidal singularities. However, whether there is any further algebro-geometric significance such as was found in [3] or whether every commutative ring of matrix ordinary differential operators with these properties is necessarily bispectral have not been considered. These questions, along with the obvious question of what other matrix differential operators satisfy equations of the form (19) will hopefully be addressed in future papers. Acknowledgements The second author was partially supported by the NSF Grant DMS0400484. The third author appreciates helpful discussions with Tom Ivey, Folkert Müller-Hoissen, and Oleg Smirnov and the support of his department during his sabbatical.

References 1. Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30(1), 95–148 (1977) 2. Ben-Zvi, D., Nevins, T.: Flows of Calogero-Moser systems. Int. Math. Res. Not. IMRN :Art. ID rnm105 38(23), (2007) 3. Berest, Y., Wilson, G.: Ideal classes of the Weyl algebra and noncommutative projective geometry. Int. Math. Res. Not. 2002(26), 1347–1396 (2002) (with an appendix by Michel Van den Bergh) 4. Bergvelt, M., ten Kroode, F.: Partitions, vertex operator constructions and multi-component KP equations. Pacific J. Math. 171(1), 23–88 (1995) 5. Boyallian, C., Liberati, J.I.: Matrix-valued bispectral operators and quasideterminants. J. Phys. A Math. Theor. 41, 365209 (2008) 6. Castro, M.M., Grünbaum, F.A.: The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: five instructive examples. Int. Math. Res. Not., pages Art. ID 47602 33 (2006) 7. Chalub, F.A.C.C., Zubelli, J.P.: Matrix bispectrality and Huygens’ principle for Dirac operators. In: Partial Differential Equations and Inverse Problems of Contemp. Math., vol. 362, pp. 89–112. American Mathematical Society, Providence (2004) 8. Dimakis, A., Müller-Hoissen, F.: With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices. J. Phys. A 40(17), F321–F329 (2007) 9. Duistermaat, J.J., Grünbaum, F.A.: Differential equations in the spectral parameter. Comm. Math. Phys. 103(2), 177–240 (1986) 10. Durán, A.J., Grünbaum, F.A.: Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. I. Constr. Approx. 22(2), 255–271 (2005) 11. Etingof, P., Varchenko, A.: Traces of intertwiners for quantum groups and difference equations. I. Duke Math. J. 104(3), 391–432 (2000) 12. Etingof, P., Schiffmann, O., Varchenko, A.: Traces of intertwiners for quantum groups and difference equations. Lett. Math. Phys. 62(2), 143–158 (2002) 13. Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in integrable systems and gauge theories. J. High Energy Phys. Paper 28, 40(7) (2000) 14. Gekhtman, M., Kasman, A.: On KP generators and the geometry of the HBDE. J. Geom. Phys. 56(2), 282–309 (2006) 15. Gibbons, J., Hermsen, T.: A generalisation of the Calogero-Moser system. Phys. D 11(3), 337– 348 (1984)

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16. Grünbaum, F.A., Iliev, P.: A noncommutative version of the bispectral problem. J. Comput. Appl. Math. 161(1), 99–118 (2003) 17. Haine, L.: KP trigonometric solitons and an adelic flag manifold. SIGMA Symmetry Integrability Geom. Methods Appl. 3:Paper 015, 15 pp. (2007) (electronic) 18. Harnad, J., Kasman, A. (eds.): The bispectral problem, volume 14 of CRM Proceedings & Lecture Notes. American Mathematical Society, Providence, RI, 1998. Papers from the CRM Workshop held at the Université de Montréal, Montreal, PQ (1997) 19. Kasman, A.: Bispectral KP solutions and linearization of Calogero-Moser particle systems. Comm. Math. Phys. 172(2), 427–448 (1995) 20. Kasman, A.: The bispectral involution as a linearizing map. In: Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Series in Mathematical Physics, pp. 221–229. Springer, New York (2000) 21. Kasman, A.: On the quantization of a self-dual integrable system. J. Phys. A 34(32), 6307–6312 (2001) 22. Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math. 31(4), 481–507 (1978) 23. Kriˇcever, I.M.: On the rational solutions of the Zaharov-Šabat equations and completely integrable systems of N particles on the line. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 84, 117–130, 312, 318 (1979) (Boundary value problems of mathematical physics and related questions in the theory of functions, 11) 24. Krichever, I.M., Babelon, O., Billey, E., Talon, M.: Spin generalization of the Calogero-Moser system and the matrix KP equation. In: Topics in Topology and Mathematical Physics, vol. 170 of Amer. Math. Soc. Transl. Ser. vol. 2, pp. 83–119. American Mathematical Society, Providence (1995) 25. Polychronakos, A.P.: The physics and mathematics of Calogero particles. J. Phys. A 39(41), 12793–12845 (2006) 26. Sakhnovich, A., Zubelli, J.P.: Bundle bispectrality for matrix differential equations. Integr. Equ. Oper. Theory 41(4), 472–496 (2001) 27. Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. In: Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), pp. 259–271. North-Holland, Amsterdam (1983) 28. Segal, G., Wilson, G.: Loop groups and equations of KdV type. Inst. Hautes Études Sci. Publ. Math. 61, 5–65 (1985) 29. Shiota, T.: Calogero-Moser hierarchy and KP hierarchy. J. Math. Phys. 35(11), 5844–5849 (1994) 30. van Diejen, J.F., Vinet, L. (eds.): Calogero-Moser-Sutherland Models. CRM Series in Mathematical Physics, New York. Springer, New York (2000) 31. Wilson, G.: Bispectral commutative ordinary differential operators. J. Reine Angew. Math. 442, 177–204 (1993) 32. Wilson, G.: Collisions of Calogero-Moser particles and an adelic Grassmannian. Invent. Math. 133(1), 1–41 (1998) (with an appendix by Macdonald, I.G.) 33. Wilson, G.: The complex Calogero-Moser and KP systems. In: Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Series in Mathematical Physics, pp. 539–548. Springer, New York (2000) 34. Zubelli, J.P.: Differential equations in the spectral parameter for matrix differential operators. Phys. D 43(2–3), 269–287 (1990)

Math Phys Anal Geom (2009) 12:201–217 DOI 10.1007/s11040-009-9060-4

Curvature Properties of Lorentzian Manifolds with Large Isometry Groups Wafaa Batat · Giovanni Calvaruso · Barbara De Leo

Received: 11 September 2008 / Accepted: 3 April 2009 / Published online: 23 April 2009 © Springer Science + Business Media B.V. 2009

Abstract The curvature of Lorentzian manifolds (Mn , g), admitting a group of isometries of dimension at least 1/2n(n − 1) + 1, is completely described. Interesting behaviours are found, in particular as concerns local symmetry, local homogeneity and conformal flatness. Keywords Lorentzian manifolds · Homogeneous spaces · Symmetric spaces · Curvature · Einstein-like metrics Mathematics Subject Classifications (2000) 53C50 · 53C20 · 53C30

1 Introduction Given a class of Lorentzian manifolds, the study of their curvature properties is an essential step for the development of our knowledge of these spaces. For example, it is needed in order to describe their parallel hypersurfaces,

Second and third authors supported by funds of MURST and the University of Salento. W. Batat Département de Mathématiques et Informatique, École Normale Superieure de L’Enseignement Technique d’Oran, Oran, Algeria e-mail: [email protected] G. Calvaruso (B) · B. De Leo Dipartimento di Matematica “E. De Giorgi”, University of Salento, Lecce, Italy e-mail: [email protected] B. De Leo e-mail: [email protected]

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or to decipher the geometric information encoded by some special curvature operators. On the one hand, this study makes possible a comparison between Riemannian results and their Lorentzian analogues, which improves our understanding of which properties are more strictly related to the signature of the metric tensor and which ones are more general. On the other hand, such a study is also motivated by the physical relevance of Lorentzian geometric models. In Riemannian geometry, extensive studies have been made about the curvature, while the corresponding study in the Lorentzian framework is comparatively recent, and many interesting cases have still to be investigated. An n-dimensional Riemannian manifold that admits a group of isometries of dimension 1/2n(n − 1) + 1, is either of constant sectional curvature or the Riemannian product between an (n − 1)-dimensional manifold of constant sectional curvature and a line (or circle). More cases are possible in Lorentzian settings, because of the existence of null submanifolds which are left invariant by group actions. Let l0 (n) > l1 (n) > ... be the possible dimensions of all groups of isometries of Lorentzian manifolds of dimension n. Following [15], an ndimensional connected Lorentzian manifold M is said to belong to the jstratum if there is a Lie group G, of dimension l j(n), that acts effectively on M by isometries. The first three strata 0, 1 and 2 correspond to Lorentzian n-manifolds admitting a group of isometries of dimension l0 (n) = 1/2n(n + 1), l1 (n) = 1/2n(n − 1) + 2 and l2 (n) = 1/2n(n − 1) + 1 respectively, and were studied in [15]. Lorentzian manifolds belonging to the first two strata are homogeneous. The complete classification of Lorentzian manifolds in the third stratum was also given, in any dimension n greater than 5 and different from 7. Besides Lorentzian manifolds of constant curvature M1n (c) and manifolds reducible as products Mn−1 (c) × R1 and R × M1n−1 (c), the remaining examples are the following ones: •

ε-spaces: Lorentzian manifolds (Rn , gε ), where ε = ±1 and  n−2  n−2   2 gε = (dxi ) − dxn−1 dxn + ε xi2 (dxn−1 )2 . i=1

•

(1.1)

i=1

ε-spaces are well known in literature. In particular, they have been proved to be irreducible Lorentzian symmetric spaces which are models for nonsymmetric spaces [4, 16]. Egorov spaces: Lorentzian manifolds (Rn , g f ), where f is a positive smooth function of a real variable and g f = f (xn )

n−2 

(dxi )2 + 2dxn−1 dxn .

(1.2)

i=1

These manifolds are named after I.P. Egorov, who first introduced and studied them in [10]. Up to our knowledge, no further investigations have been made about Lorentzian metrics described in (1.2). It is then worthwhile to investigate

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the curvature properties of this family of metrics and this is the aim of the present paper. The paper is organized in the following way. In Section 2, the Levi Civita connection, the curvature tensor and the Ricci tensor of (Rn , g f ) will be described in terms of components with respect to coordinate vector fields {∂i = ∂∂x }. This provides the needed information for the study of the geometry i of (Rn , g f ), which we make in Sections 3 and 4. In particular, we prove in Section 3 that all Egorov spaces are geodesically complete, admit a parallel null vector field and are conformally flat. Examples featuring some important curvature properties as local symmetry and the Einstein condition are also completely characterized. In Section 4, homogeneity of (Rn , g f ) will be discussed, via the existence of a Lorentzian homogeneous structure. We first deal with the three-dimensional case, where we also obtain the full classification of Lorentzian homogeneous structures on (M3 , g f ). We then characterize reductive homogeneous Egorov spaces in all dimensions. It turns out that, differently from all the other Lorentzian manifolds in the third stratum, homogeneity is very rare for Egorov spaces. Finally, in Section 5 we obtain some results concerning the curvature of an ε-space and we conclude with some general remarks on the geometry of Lorentzian manifolds in the third stratum.

2 Connection and Curvature of Egorov Spaces We denote by ∇ the Levi Civita connection of (Rn , g f ) and by R its curvature tensor, taken with the sign convention: R(X, Y) = ∇[X,Y] − [∇ X , ∇Y ].

(2.1)

If not stated otherwise, throughout the paper we shall assume n  3. With respect to the basis of coordinate vector fields {∂i = ∂∂x } for which (1.2) holds, i the metric components are gij = δij f

for i  n − 2 and j  n,

gn−1n−1 = gnn = 0,

gnn−1 = gn−1n = 1.

(2.2)

By (2.2), a standard calculation gives the Christoffel symbols ijk . The only possible non-vanishing Christoffel symbols are the following ones: iin−1 = −

f , 2

i in =

f , 2f

i = 1, . . . , n − 2.

(2.3)

Hence, the possibly non-vanishing covariant derivatives of coordinates vector fields are given by ∇∂i ∂i = −

f ∂n−1 , 2

∇∂i ∂n =

f ∂i , 2f

i = 1, . . . , n − 2.

(2.4)

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Using (2.4) in (2.1), we get that the only possible non-vanishing curvature components are n−1 = Rini

 1   2 ( f ) − 2 f f  , 4f

Riinn = −

 1   2 ( f ) − 2 f f  , 2 4f

i = 1, . . . , n − 2,

(2.5)

and the ones obtained by them using the symmetries of the curvature tensor. Consequently, the components of the (0, 4)-curvature tensor R(X, Y, Z , W) = g(R(X, Y)Z , W), with respect to {∂i }, are given by    Rinin = 41f ( f  )2 − 2 f f  , i = 1, . . . , n − 2, (2.6) Rijkh = 0 otherwise. We can now calculate the components of the covariant derivative ∇ R with respect to {∂i }. By (2.4) and (2.5), we easily get ⎧   ⎪ (∇ R)(∂i , ∂n , ∂n ) = 2 1f 3 ( f  )3 + f 2 f  − 2 f f  f  ∂i , i = 1, . . . , n − 2, ⎪ ⎨ ∂n   (∇∂n R)(∂i , ∂n , ∂i ) = − 2 1f 2 ( f  )3 + f 2 f  − 2 f f  f  ∂n−1 , i = 1, . . . , n − 2, ⎪ ⎪ ⎩ (∇ R)(∂ , ∂ , ∂ ) = 0 otherwise. ∂i

j

k

h

(2.7)  3







 2



ff . Note that ( f ) + f 2ff 3−2 f f f is nothing but the first derivative of − ( f )4−2 f2 Next, as it is well known, given a pseudo-Riemannian manifold (Mn , g), its Ricci tensor  is defined as the contraction of the curvature tensor: 2

(X, Y) = tr(R(X, ·, Y, ·))

(2.8)

and its components with respect to a basis of coordinate vector fields are given by  m Rimj . ij = (∂i , ∂ j) = In our case, by (2.5) we then obtain nn =

n−2 [( f  )2 − 2 f f  ], 4 f2

ij = 0

otherwise.

(2.9)

Moreover, (2.4) and (2.9) easily give the components of the covariant derivative of : ∇n nn = −

n−2 [( f  )3 + f 2 f  − 2 f f  f  ], 2 f3

∇i  jk = 0

otherwise.

(2.10)

3 The Geometry of Egorov Spaces (Rn , g f ) In the previous Section, we completely described the Levi Civita connection, the curvature tensor and the Ricci tensor of the Lorentzian manifold (Rn , g f ).

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This provides all needed information in order to investigate the geometry of (Rn , g f ). We consider first geometrical properties determined by the Levi Civita connection, starting from geodesic completeness. Let γ (t) be a geodesic curve on (Rn , g f ), determined by its components (γ1 (t), .., γn (t)) with respect to {∂i }. From the geodesic equation ∂ 2 γk  k ∂γi ∂γ j + ij (γ (t)) =0 ∂t2 ∂t ∂t i, j=1 n

using (2.3) we get the following system of partial differential equations: ⎧ 2 ∂ γk f ∂γk ∂γn ⎪ ⎪ ⎪ + (γn (t)) = 0, k = 1, .., n − 2, ⎪ 2 ⎪ ∂t 2f ∂t ∂t ⎪ ⎪

  ⎨ ∂ 2γ 1  ∂γ1 2 ∂γn−2 2 n−1 (3.1) = 0, − f (γn (t)) + .. + ⎪ ∂t2 2 ∂t ∂t ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ∂ γn = 0. ∂t2 Integrating twice the last equation in (3.1), we get γn (t) = an t + bn , where an , bn are real constants. We now replace γn (t) = an t + bn into the remaining equations of (3.1) and integrate twice. If an = 0, we easily obtain ⎧ ⎪ ⎨ γk (t) = ak t + bk , k = 1, .., n − 2, γn−1 (t) = 12 (a21 + .. + a2k−2 ) f  (bn )t + bn−1 , ⎪ ⎩ γn (t) = bn ,

(3.2)

where bn−1 and ak , bk , k = 1, .., n − 2, are real constants. When an = 0, we get ⎧  −1 ⎪ ⎪ γk (t) = rk [ f (an t + bn )] 2 dt, k = 1, .., n − 2, ⎨ r2 +..+r2  (3.3) γn−1 (t) = 1 2an k−2 ln[ f (an t + bn )]dt + rn−1 t, ⎪ ⎪ ⎩ γn (t) = an t + bn , where r1 , .., rn−1 are real constants. All curves having components given by either (3.2) or (3.3) are defined on the entire real line (in the second case, taking into account the fact that f is a positive smooth function). Hence, we proved the following Theorem 3.1 All Egorov spaces (Rn , g f ) are geodesically complete. The next remark concerns the existence of parallel null vector fields. Since (Rn , g f ) is irreducible [15], it does not admit an either spacelike or timelike parallel vector field. However, by (2.2) it follows that ∂n−1 is a null vector field and, by (2.4), ∇∂i ∂n−1 = 0 for all i = 1, . . . , n. Therefore, we have the following

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Theorem 3.2 All Egorov spaces (Rn , g f ) admit a parallel null vector field. A complete local description of Lorentzian three-manifolds admitting a parallel null vector field was given in [8]. In particular, the Ricci operator g(QX, Y) = (X, Y) of such a space is never diagonalizable, except for the trivial flat case [6, 8]. As it is well known, the fact that in Lorentzian geometry a self-adjoint operator needs not to be diagonalizable, is the cause of several important differences between Riemannian and Lorentzian geometries [14]. It is easy to check that the same phenomenon occurs for the Ricci operator of (Rn , g f ). In fact, by (2.2) and (2.9) we easily get that, with respect to {∂i }, the Ricci operator is described by ⎛

0 ··· 0 ⎜ .. .. .. ⎜ Q=⎜. . . ⎝0 ··· 0 0 ··· 0

0 .. .

⎞

⎟ ⎟ ⎟. n−2  2  ⎠ [( f ) − 2 f f ] 2

(3.4)

4f

0

By (3.4), the Ricci operator Q is nilpotent and λ = 0 is the only Ricci eigenvalue, of multiplicity n. The corresponding eigenspace is (n − 1)-dimensional, unless ( f  )2 − 2 f f  = 0 which, by (2.5), implies that R = 0, that is, (Rn , g f ) is flat. Therefore, excluding the flat case, the Ricci operator of an Egorov space is never diagonalizable. Since all Ricci eigenvalues vanish, we also have at once that the scalar curvature of (Rn , g f ) vanishes. So, we proved the following Theorem 3.3 The only Ricci eigenvalue of any Egorov space (Rn , g f ) is λ = 0. In particular, the scalar curvature of (Rn , g f ) vanishes identically. Unless (Rn , g f ) is flat, its Ricci operator Q is nilpotent. Formulas (2.2) and (2.9) imply at once that if an Egorov space (Rn , g f ) is Einstein, then ( f  )2 − 2 f f  = 0

(3.5)

and so, the manifold is flat. Note also that since the scalar curvature vanishes, (Rn , g f ) is Einstein if and only if it is Ricci-flat. By solving differential equation (3.5) explicitly, we obtain the following Theorem 3.4 Let (Rn , g f ) be an Egorov space. Then, the following conditions are equivalent: (i) (ii) (iii) (iv)

(Rn , g f ) is Einstein, (Rn , g f ) is Ricci-flat, (Rn , g f ) is flat, either the function f > 0 is a constant, or f (xn ) = a(xn + b )2 , for two real constants a > 0 and b .

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Next, we recall that a pseudo-Riemannian manifold (Mn , g), of dimension n  4, is conformally flat if and only if its Weyl curvature tensor vanishes, that is, R(X, Y, Z , W) =

1 (g(X, Z )(Y, W) + g(Y, W)(X, Z ) − n−2 −g(X, W)(Y, Z ) − g(Y, Z )(X, W)) − τ (g(X, Z )g(Y, W) − g(Y, Z )g(X, W)), − (n − 1)(n − 2) (3.6)

for all vector fields X, Y, Z , W tangent to M, where τ denotes the scalar curvature. By (3.6), the curvature of a conformally flat manifold is completely determined by its Ricci tensor. Therefore, besides spaces of constant curvature, conformally flat manifolds are the spaces with the simplest possible curvature. In Riemannian geometry, conformal flatness has been intensively studied, but it only gives few examples under homogeneity conditions. In fact, a locally symmetric conformally flat Riemannian manifold is either of constant sectional curvature, or locally isometric to a Riemannian product R × Sn−1 (k), R × Hn−1 (−k), S p (k) × Hn− p (−k) [17]. These also are the only conformally flat locally homogeneous Riemannian manifolds [18] and even the weaker assumption of curvature homogeneity does not lead to other examples [7]. In the case of an Egorov space (Rn , g f ), we have τ = 0 and by (2.6) and (2.9) it easily follows that (3.6) holds for all coordinate vector fields. Therefore, the Weyl tensor of (Rn , g f ) vanishes, that is, Theorem 3.5 All Egorov spaces (Rn , g f ), n  3, are conformally flat. In the statement of Theorem 3.5 we also included the three-dimensional case, for which the conformal flatness condition is given by the vanishing of the Schouten tensor c, that is, c(X, Y, Z ) = (∇ X )(Y, Z ) − (∇Y )(X, Z ) − 1/2 (g((∇ X τ )Y, Z )− −g((∇Y τ )X, Z )) = 0.

(3.7)

For an Egorov space (Rn , g f ), of any dimension n  3, equation (3.7) follows from the generally stronger condition that the Ricci tensor is Codazzi, that is, (∇ X )(Y, Z ) = (∇Y )(X, Z ).

(3.8)

Condition (3.8), defining one class of the so called Einstein-like metrics, was first introduced and studied in the Riemannian framework [12]. Recently,

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Einstein-like pseudo-Riemannian metrics have been extensively studied (see for example [6, 8]). A pseudo-Riemannian manifold whose Ricci tensor satisfies (3.8) is also said to have harmonic curvature (see for example [9]). In terms of local components, (3.8) is equivalent to requiring that ∇i  jk = ∇ jik ,

for all i, j, k.

(3.9)

But (2.10) implies at once that (3.9) holds. Hence, we have the following Theorem 3.6 All Egorov spaces (Rn , g f ), n  3, have harmonic curvature. We now recall that a symmetric space is a connected pseudo-Riemannian manifold whose geodesic symmetries are isometries. When (M, g) is locally isometric to a symmetric space, it is called locally symmetric. The well known characterization states that this is equivalent to requiring that ∇ R = 0. Since each Egorov space is conformally flat, ∇ R = 0 is equivalent to ∇ = 0. By (2.10) we see at once that an Egorov space (Rn , g f ) is Ricci-parallel if and only if

 2  ( f  )3 + f 2 f  − 2 f f  f  ( f ) − 2 f f  =− = 0, 2 f3 4 f2 that is, ( f  )2 − 2 f f  = kf 2 ,

(3.10)

where k is a real constant. So, we obtained the following Theorem 3.7 Let (Rn , g f ) be an Egorov space. Then, the following conditions are equivalent: • • •

(Rn , g f ) is locally symmetric, (Rn , g f ) is Ricci-parallel, the function f is a solution of (3.10).

Note that if k = 0, then (3.10) reduces to the flatness condition ( f  )2 − 2 f f  = 0, which we already integrated. When k < 0, explicit solutions of (3.10) are given by f (xn ) = μe±

√

−kxn

,

(3.11)

for any real constant μ > 0.

4 Homogeneity of Egorov Spaces A pseudo-Riemannian manifold (M, g), having a large group of isometries, should have a good chance to be homogeneous. Patrangenaru [15] proved

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directly that in any dimension n  3, Lorentzian manifolds in the first two strata are homogeneous. Indeed, he also proved a stronger condition, by showing that Lorentzian manifolds in the first two strata have constant sectional curvature. Moreover, the classification of Lorentzian manifolds in the third stratum (in dimension n > 5, n = 7) shows that, apart from Egorov spaces, all the remaining examples are symmetric spaces. In this context, Egorov spaces already proved to be an exception, since Theorem 3.7 shows that they are symmetric only under a very restrictive condition for the defining function f . It is then natural to ask the following Question are Egorov spaces (Rn , g f ) homogeneous? As we shall see, the answer is negative for almost any defining function f . To prove this fact, we shall make use of the notion of homogeneous structure. Homogeneous pseudo-Riemannian structures were introduced by Gadea and Oubiña in [13], in order to obtain a characterization of reductive homogeneous pseudo-Riemannian manifolds, similar to the one given for homogeneous Riemannian manifolds by Ambrose and Singer [1] (see also [19]). Definition 4.1 [13] A homogeneous pseudo-Riemannian structure on a pseudoRiemannian manifold (M, g) is a tensor field T of type (1, 2) on M, such that the connection ∇˜ = ∇ − T satisfies ˜ = 0, ∇g

∇˜ R = 0,

˜ = 0. ∇T

More explicitly, T is the solution of the following system of equations (known as Ambrose-Singer equations): g(T X Y, Z ) + g(Y, T X Z ) = 0,

(4.1)

(∇ X R)Y Z = [T X , RY Z ] − RT X Y Z − RYT X Z ,

(4.2)

(∇ X T)Y = [T X , TY ] − TT X Y ,

(4.3)

for all vector fields X, Y, Z . The geometric meaning of the existence of a homogeneous pseudo-Riemannian structure is explained by the following Theorem 4.2 [13] A connected, simply connected and complete pseudoRiemannian manifold (M, g) admits a pseudo-Riemannian structure if and only if it is a reductive homogeneous pseudo-Riemannian manifold. If at least one of the hypotheses of connectedness, simply connectedness or completeness is lacking, the existence of a solution of (4.1)–(4.3) implies that

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(M, g) is locally isometric to a reductive homogeneous space (and so, locally homogeneous). As we already remarked, Lorentzian manifolds in the first three strata, except for Egorov spaces, are symmetric. In particular, they are reductive homogeneous spaces. We can now prove the following Theorem 4.3 Necessary condition for an Egorov space (Rn , g f ) to be locally isometric to a reductive homogeneous space, is that its defining function satisfies one of the following conditions: • •

either f is a solution of (3.10) (locally symmetric case), or f is a solution of ( f  )2 − 2 f f  =

cn f 2, (xn + d)2

(4.4)

for some real constants cn = 0 and d. Proof If (Rn , g f ) is locally isometric to a reductive homogeneous space, then it admits a homogeneous Lorentzian structure T. This tensor is uniquely determined by its components Tijk with respect to the coordinate vector fields  {∂i }, defined by T(∂i , ∂ j) = k Tijk ∂k . Rewriting (4.1)–(4.3) for the coordinate vector fields ∂i , we obtain r Tijr grk + Tik grj = 0,

(4.5)

s ∇i  jk = −Tijr rk − Tik  js ,

(4.6)

(∇i T)∂ j = T∂i T∂ j − T∂ j T∂i − TT∂i ∂ j ,

(4.7)

for all indices i, j, k. Note that (4.2) has been replaced by the condition (4.6) about the Ricci tensor, since (Rn , g f ) is conformally flat. It is not an easy task to solve (4.5)–(4.7) in full generality. However, from (4.5) with i = j = k = n−1 n we have Tnn = 0. We then calculate (4.6) for i = j = k = n, and (4.7) for (i, j) = (n, n) and applied to ∂n . Taking into account the fact that nn = nn (xn )  and ∇n nn = nn , we respectively get n = h(xn ) = − Tnn

∇n nn  = − nn 2nn 2nn

n n 2 ∂n Tnn = −(Tnn ) ,

and

(4.8) (4.9)

where we assumed nn = 0, thus excluding the trivial flat case. Therefore, one of the following cases occurs: n FIRST CASE: Tnn = 0. Then, ∇n nn = 0. By (2.10) it follows that (Rn , g f ) is Ricci-parallel and so, locally symmetric. In particular, by

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Theorem 3.7 it follows that f satisfies (3.10). In this case, T = 0 is a homogeneous structure on (Rn , g f ). n = 0. We use (4.8) in (4.9), obtaining that nn = nn (xn ) SECOND CASE: Tnn must be a solution of differential equation  2  3(nn ) − 2nn nn = 0.

(4.10)

α Therefore, (4.10) implies that either nn is a constant, or nn = (xn +d) 2 , for two real constants α = 0 and d. If nn is constant then by (2.10) we get ∇n nn = 0, α n = 0. So, nn = (xn +d) which, by (4.8), contradicts Tnn 2 . Then, using (2.9) to write 4α nn in function of f and putting cn = n−2 = 0, (4.10) becomes (4.4) and this ends the proof  

To prove the converse of Theorem 4.3, we must show that whenever the defining function f satisfies (4.4), the Egorov space (Rn , g f ) admits a homogeneous structure, that is, the system (4.5)–(4.7) admits a solution. In order to envisage how to determine such a solution, we first solve the system (4.5)–(4.7) in dimension three, by proving the following Theorem 4.4 An Egorov space (R3 , g f ) is locally homogeneous if and only if either f is a solution of (3.10) (locally symmetric case), or f is a solution of (4.4) for n = 3 (non locally symmetric case). Proof The “only if” part follows at once from Theorem 4.3. For the “if” part, we first recall that a three-dimensional homogeneous Lorentzian manifold is necessarily reductive. This was proved in [11] and also follows independently from the classification the second author gave in [5]. So, if (R3 , g f ) is homogeneous, then it must admit a homogeneous Lorentzian structure T. By Theorem 4.3 we know that the defining function f satisfies either (3.10) or (4.4). In the first case, (R3 , g f ) is locally symmetric. Hence, in order to complete the proof, we now assume that f > 0 is a solution of (4.4) for n = 3. k From (4.5)–(4.6) we easily get the following algebraic restrictions on Tik : ⎧ 1 Ti1 = 0 for all i = 1, 2, 3, ⎪ ⎪ ⎪ ⎪ ⎪ 3 2 ⎪ ⎪ ⎨ Ti2 = Ti3 = 0 for i = 1, 2, 2 1 T11 = − f T13 , ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ 3 2 ⎪ = −T32 = − 33 . ⎩ T33 233

(4.11)

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1 1 1 So, in order to determine T, we must determine T13 , T23 and T33 . Denoting 3 by (x, y, z) the coordinates in R , we find that because of (4.7), these functions satisfy the following system of partial differential equations: ⎧ 1 1 1 1 ⎪ ∂x T13 = f T23 T13 − 1/2 f  T23 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 2 ⎪ ⎪ = f (T23 ) , ∂ y T13 ⎪ ⎪ ⎪  ⎪ ⎪  ⎪ 1 1 1 1 ⎪ ∂z T13 = 33 T13 + f T23 T33 , ⎪ ⎪ 2 ⎪ 33 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ∂x T23 = 0, ⎪ ⎪ ⎨ 1 ∂ y T23 = 0, (4.12) ⎪ 1  1 ⎪ 1 ⎪ ∂z T23 = − f T23 , ⎪ ⎪ ⎪ ⎪

2f  ⎪   ⎪ 33 1  ⎪ 1 1 1 ⎪ T = − T + f ∂ T , ⎪ x 33 13 13 ⎪ ⎪ 2f 233 ⎪ ⎪ ⎪  ⎪ 1 1 1 1 ⎪ = − 33 T23 − T13 T23 , ∂ y T33 ⎪ ⎪ ⎪ 2 33 ⎪ ⎪   ⎪ 33 1  ⎪ 1 1 1 ⎪ f ∂ T = −T + T − . ⎩ z 33 13 33 2f 233

Next, we derive the first equation of (4.12) by y and the second equation of (4.12) by x, obtaining respectively 1 1 1 = f T23 ∂ y T13 , ∂ y ∂x T13

1 ∂x ∂ y T13 = 0.

1 1 Therefore, f T23 ∂ y T13 = 0, which, taking into account the second equation of 1 (4.12), gives T23 = 0. Hence, (4.12) reduces to ⎧ 1 ∂x T13 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ∂ y T13 = 0, ⎪ ⎪ ⎪  ⎪ 33 ⎪ 1 ⎪ ⎪ ∂ T = T1 , z ⎪ 13 ⎨ 233 13 

  33 1  (4.13) 1 1 1 ⎪ T13 + , ∂x T33 = f − T13 ⎪ ⎪ 2f 233 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ∂ T = 0, y 33 ⎪ ⎪ ⎪

⎪   ⎪ 33 1  ⎪ 1 1 1 ⎪ f + T13 − . ⎩ ∂z T33 = −T33 2f 233 1 only depends on x3 = z. The first two equations in (4.13) yield that T13 α Since f satisfies (4.4), by (4.10) we have that 33 = (z+d) 2 , with α  = 0. The third equation in (4.13) then becomes 1 ∂z T13 =−

1 T1 . z + d 13

(4.14)

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μ 1 Integrating (4.14), we easily find T13 = z+d , for a real constant μ (also including 1 the case μ = 0, which implies T13 = 0). Next, we integrate the fourth equation in (4.13), where the second member μ α 1 = z+d and 33 = (z+d) does not depend on x. Also using T13 2 , we get

 μ μ − 1 f 1 T33 = − x + p(z), (4.15) z+d 2f z+d

where p = p(z) is a smooth function. We then replace (4.15) in the last equation of (4.13), obtaining    

 μ − 1 f f  − ( f  )2 μ f μ μ−1 + − x+ − z+d 2 f2 (z + d)2 2f z + d (z + d)2

   μ − 1 f μ+1 f μ  + p (z) = − + − x− z+d 2f z+d 2f z+d

  f μ+1 − + p(z). (4.16) 2f z+d Since (4.16) holds for all x, it yields the following two conditions:    ⎧ ⎨ p (z) + f + μ+1 p(z) = 0, 2f z+d   ⎩ μ−1 2 f f  −(2 f  )2 − μ(μ−1)2 = 0. z+d 4f (z+d) 

(4.17)

 2

f) We can calculate 2 f f 4−( by (4.4). Taking into account c3 = 4α, the second f2 equation of (4.17) then reduces to

−

(μ − 1)(μ2 − μ + α) = 0, (z + d)3 √

for all z, that is, either μ = 1 or μ = 1± 21−4α (but only when α  14 ). In both cases, we are left with the first equation of (4.17), whose explicit solutions are given by p(z) =

β (z + d)μ+1

 , f

for any real constant β. Therefore, we proved that ⎧ √ ⎪ ⎪ T 1 = μ , either μ = 1 or μ = 1 ± 1 − 4α and α  1/4, ⎪ ⎪ ⎨ 13 z+d 2 1 T23 = 0,

 ⎪ ⎪ μ − 1 f μ β ⎪ 1 ⎪ T33  , = − x+ ⎩ z+d 2f z+d (z + d)μ+1 f

(4.18)

satisfy (4.12) and so, together with (4.11), they determine a homogeneous structure on (R3 , g f ). Hence, the space is locally homogeneous  

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Notice that in the proof of Theorem 4.4, we completely solved the system (4.5)–(4.7) under the assumption that f satisfies (4.4). Thus, we proved the following Corollary 4.5 Let (R3 , g f ) be a nonsymmetric locally homogeneous Egorov space. Then, all and the ones homogeneous structures on (R3 , g f ) are determined by (4.11) and (4.18). The complete solution of the three-dimensional case suggests to us the simplest form for a homogeneous structure T on (Rn , g f ), with f satisfying (4.4). In dimension three, this homogeneous structure is the one determined by (4.11) and (4.18) with μ = 1 and β = 0, that is, having 1 3 2 T13 = T33 = −T32 =

1 , z+d

2 T11 =−

1 f z+d

as the only non-vanishing components. The generalization of this solution of (4.5)–(4.7) to the n-dimensional case, is the homogeneous structure T, whose only non-vanishing components, with respect to the local coordinate basis {∂1 , .., ∂n }, are given by n−1 i n Tin = Tnn = −Tnn−1 =

1 , xn + d

Tiin−1 = −

1 f, xn + d

for all i  n − 2.

(4.19) In fact, long but routine calculations show that if (4.19) holds, then equations (4.5)–(4.7) are satisfied. Hence, we obtained the following complete characterization: Theorem 4.6 An Egorov space (Rn , g f ) is locally isometric to a reductive homogeneous space if and only if the defining function f is a solution of either (3.10) (locally symmetric case) or (4.4) (non-symmetric case). Proof The “only if” part was proved in Theorem 4.3. For the “if” part, if f satisfies (3.10), then (Rn , g f ) is locally symmetric (Theorem 3.7). On the other hand, if f satisfies (4.4), then (4.19) determines a homogeneous structure on  (Rn , g f ). So, (Rn , g f ) is locally isometric to a reductive homogeneous space  Remark 4.7 Curvature homogeneity. A pseudo-Riemannian manifold (M, g) is called curvature homogeneous if, for any points p, q ∈ M, there exists a linear isometry φ : T p M → Tq M such that φ ∗ (R(q)) = R( p). Locally homogeneous spaces are curvature homogeneous, but in any dimension n  3, several examples are known of curvature homogeneous spaces which are not locally homogeneous. We can refer to [2] for a survey and further references for the Riemannian case, while the first Lorentzian examples were given in [3]. So, given a class of pseudo-Riemannian metrics, in general one can expect to find many more curvature homogeneous than locally homogeneous examples. As concerns Egorov spaces (Rn , g f ), this is not an easy problem.

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By the definition it follows that a curvature homogeneous space admits a (local) pseudo-orthonormal frame field, with respect to which all curvature components are constant. However, it is not easy to determine this frame field, starting from a description of the manifold like the one we obtained in Section 2 for Egorov spaces. Notice that the constancy of the scalar curvature does not give restrictions, since we proved in Theorem 3.3 that the scalar curvature identically vanishes on all Egorov spaces. Finally, using (2.2) we can easily construct a particular pseudo-orthonormal frame field on (Rn , g f ), namely, ei = f −1/2 ∂i,

i = 1, .., n − 2,

1 en−1 = √ (∂n−1 + ∂n ), 2

1 en = √ (∂n−1 − ∂n ). 2 By (3.4) we get that, with respect to this pseudo-orthonormal frame field, the Ricci operator is given by ⎛ ⎞ 0 ··· 0 0 .. .. ⎜ .. .. ⎟ ⎜. . ⎟ . . ⎟ ⎜ n−2 n−2  2   2  ⎟ Q=⎜ [( f ) − 2 f f ] − [( f ) − 2 f f ] ⎜0 ··· ⎟ ⎟ ⎜ 8 f2 8 f2 ⎝ ⎠ n−2 n−2  2   2  0 ··· [( f ) − 2 f f ] − [( f ) − 2 f f ] 8 f2 8 f2 and so, if we require that all components of Q with respect to {ei } are constant, we obtain equation (3.10), which characterizes locally symmetric Egorov spaces. For all these reasons, it is an interesting open problem either to find curvature homogeneous Egorov spaces which are not locally homogeneous, or to prove that a curvature homogeneous Egorov space is necessarily locally homogeneous.

5 Curvature of (Rn , gε ) Lorentzian spaces (Rn , gε ) are symmetric and their curvature was described in [4]. However, we add here further information about the curvature and related geometry of these spaces, also to get some general conclusions about Lorentzian manifolds in the third stratum. Calculating the Christoffel symbols with respect to the local coordinate vector fields for which (1.1) holds, we easily find that the Levi Civita connection of (Rn , gε ) is completely determined by ∇∂i ∂n−1 = −2εxi ∂n ,

i = 1, . . . , n − 2,

∇∂n−1 ∂n−1 = −ε

n−2  k=1

xk ∂k , (5.1)

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while the remaining covariant derivatives vanish identically. In particular, ∇∂i ∂n = 0 for all indices i. By (1.1) it follows that ∂n is a null vector field. Hence, we proved the following Theorem 5.1 (Rn , gε ) admits a parallel null vector field. Next, by (5.1) and (2.1), we get the curvature components. For the (0, 4)curvature tensor, they are given by Rin−1in−1 = −ε,

i = 1, . . . , n − 2,

Rijkh = 0 otherwise.

(5.2)

As a consequence, the components of the Ricci tensor with respect to the basis of coordinate vector fields are given by n−1n−1 = −(n − 2)ε,

ij = 0

otherwise.

(5.3)

It is then easy to check, as we did for Egorov spaces, that the Ricci operator Q of (Rn , gε ) is nilpotent and admits λ = 0 as the only eigenvalue, associated to an (n − 1)-dimensional eigenspace. Moreover, the scalar curvature τ of (Rn , gε ) vanishes and (5.2), (5.3) easily imply that (3.6) holds for all coordinate vector fields. Therefore, we have the following Theorem 5.2 (Rn , gε ) is conformally flat (and has harmonic curvature). Its only Ricci eigenvalue is λ = 0, and its Ricci operator Q is not diagonalizable and nilpotent. As we already mentioned in the Introduction, together with Lorentzian spaces of constant curvature and Lorentzian products of a space of constant curvature and a real line (or circle), Egorov spaces and ε-spaces complete the classification of Lorentzian manifolds in the third stratum, in all dimensions n > 5, n = 7. The so called h-triple method, used in [15] to obtain the classification, does not apply for n = 7. This classification result, together with Theorems (3.5) and (5.2), lead to the following Theorem 5.3 For any dimension n > 5, n = 7, Lorentzian manifolds (Mn , g) in the third stratum are conformally flat. Moreover, they have harmonic curvature. Acknowledgement The Authors wish to express their gratitude toward the Referees for their valuable remarks and suggestions.

References 1. Ambrose, W., Singer, I.M.: On homogeneous Riemannian manifolds. Duke Math. J. 25, 647–669 (1958) 2. Boeckx, E., Kowalski O., Vanhecke, L.: Riemannian Manifolds of Conullity Two. World Scientific, Singapore (1996) 3. Bueken, P., Vanhecke, L.: Examples of curvature homogeneous Lorentz metrics. Class. Quantum Gravity 14, L93–96 (1997)

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4. Cahen, M., Leroy, J., Parker, M., Tricerri, F., Vanhecke, L.: Lorentz manifolds modelled on a Lorentz symmetric space. J. Geom. Phys. 7(4), 571–581 (1990) 5. Calvaruso, G.: Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57, 1279–1291 (2007) 6. Calvaruso, G.: Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geom. Dedicata. 127, 99–119 (2007) 7. Calvaruso, G., Vanhecke, L.: Special ball-homogeneous spaces. Z. Anal. ihre Anwend. 16, 789–800 (1997) 8. Chaichi, M., García-Río, E., Vázquez-Abal, M.E.: Three-dimensional Lorentz manifolds admitting a parallel null vector field. J. Phys. A Math. Gen. 38, 841–850 (2005) 9. Derdzinski, A.: Classification of certain compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor. Math. Z. 172, 273–280 (1980) 10. Egorov, I.P.: Riemannian spaces of the first three lacunary types in the geometric sense (Russian). Dokl. Akad. Nauk. SSSR 150, 730–732 (1963) 11. Fels, M.E., Renner, A.G.: Non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Can. J. Math. 58(2), 282–311 (2006) 12. Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata. 7, 259–280 (1978) 13. Gadea, P.M., Oubina, J.A.: Homogeneous pseudo-Riemannian structrues and homogeneous almost para-Hermitian structures. Houston J. Math. 18(3), 449–465 (1992) 14. O’Neill B.: Semi-Riemannian Geometry. Academic, New York (1983) 15. Patrangenaru, V.: Lorentz manifolds with the three largest degrees of symmetry. Geom. Dedicata. 102 , 25–33 (2003) 16. Patrangenaru, V.: Locally homogeneous pseudo-Riemannian manifolds. J. Geom. Phys. 17(1), 59–72 (1995) 17. Ryan, P.: A note on conformally flat spaces with constant scalar curvature. In: Proc. 13th Biennal Seminar of the Canadian Math. Congress Differ. Geom. Appl. vol.2, pp. 115–124. Dalhousie Univ. Halifax 1971. (1972) 18. Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tohôku Math. J. 27, 103–110 (1975) 19. Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds, Notes 83. London Math. Soc. Lect. Cambridge University Press, Cambridge (1983)

Math Phys Anal Geom (2009) 12:219–254 DOI 10.1007/s11040-009-9059-x

Continuity of the Integrated Density of States on Random Length Metric Graphs Daniel Lenz · Norbert Peyerimhoff · Olaf Post · Ivan Veseli´c

Received: 3 December 2008 / Accepted: 25 March 2009 / Published online: 12 May 2009 © Springer Science + Business Media B.V. 2009

Abstract We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly

D. Lenz Fakultät für Mathematik & Informatik, Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany e-mail: [email protected] URL: http://www.analysis-lenz.uni-jena.de/ N. Peyerimhoff Department of Mathematical Sciences, Durham University, Science Laboratories South Road, Durham DH1 3LE, UK e-mail: [email protected] URL: www.maths.dur.ac.uk/˜dma0np/ O. Post (B) Institut für Mathematik, SFB 647 ‘Space – Time – Matter’, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany e-mail: [email protected] URL: www.math.hu-berlin.de/˜post/ I. Veseli´c Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany & Emmy-Noether Programme of the DFG URL: www.tu-chemnitz.de/mathematik/schroedinger/members.php

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supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states. Keywords Integrated density of states · Periodic and random operators · Metric graphs · Quantum graphs · Continuity properties Mathematics Subject Classifications (2000) 35J10 · 82B44

1 Introduction Quantum graphs are Laplace or Schrödinger operators on metric graphs. As structures intermediate between discrete and continuum objects they have received quite some attention in recent years in mathematics, physics and material sciences, see e.g. the recent proceeding volume [15] for an overview. Here, we study periodic and random quantum graphs. Our results concern spectral properties which are related to the integrated density of states (IDS), sometimes called spectral distribution function. As in the case of random Schrödinger operators in Euclidean space, disorder may enter the operator via the potential. Moreover, and this is specific to quantum graphs, randomness may also influence the characteristic geometric ingredients determining the operator, viz. • •

the lengths of the edges of the metric graph and the vertex conditions at each junction between the edges.

In the present paper we pay special attention to randomness in these geometric data. Our results may be summarised as follows. For quite wide classes of quantum graphs we establish • • •

the existence, respectively the convergence in the macroscopic limit, of the integrated density of states under suitable ergodicity and amenability conditions (see Theorem 2.6), a trace per unit volume formula for the IDS (see (2.9)), a Wegner estimate for random edge length models (assuming independence and smoothness for the disorder) (Theorem 2.9). This implies quantitative continuity estimates for the IDS (Corollary 2.10).

These abstract results are illustrated by the thorough discussion of an example concerning a combinatorial and a metric graph based on the Kagome lattice. In this case we calculate positions and sizes of all jumps of the IDS. Our results show the effect of smoothing of the IDS via randomness. The article is organised as follows: In the remainder of this section we summarise the origin of results about the construction of the IDS and of Wegner estimates and point out aspects of the proofs which are different in

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the case of quantum graphs in comparison to random Schrödinger operators on L2 (Rd ) or 2 (Zd ). We mention briefly recent results about spectral properties of random quantum graphs which are in some sense complementary to ours. Finally, we point out some open problems in this field of research. In the next section, we introduce the random length model and state the main results. In Section 3 we present the Kagome lattice example. In Section 4 we prove Theorem 2.6 concerning the approximability of the IDS. Finally, in Section 5 we prove the Wegner estimate Theorem 2.9. Intuitively, the IDS concerns the number of quantum states per unit volume below a prescribed energy. From the physics point of view the natural definition of this quantity is via a macroscopic limit. This amounts to approximating the (ensemble-averaged) spectral distribution function of an operator on the whole space by normalised eigenvalue counting functions associated to finitevolume restrictions of the operator. For ergodic random and almost-periodic operators in Euclidean space this approach has been implemented rigorously in [58, 62], and developed further in a number of papers, among them [35, 55] and [25]. All these operators were stationary and ergodic with respect to a commutative group of translations. For graphs and manifolds beyond Euclidean space the relevant group is in general no longer abelian. The first result establishing the approximability of the IDS of a periodic Schrödinger operator on a manifold was [2]. An important assumption on the underlying geometry is amenability. Analogous results on transitive graphs have been established e.g. in [57] and [56]. For Schrödinger operators with a random potential on a manifold with an amenable covering group the existence of the IDS was established in [61], and for Laplace-Beltrami operators with random metrics in [52]. For analogous results for discrete operators on amenable graphs see e.g. [66] and [54]. A key ingredient of the proofs of the above results is the amenable ergodic theorem of [48]. More recently, the question of approximation of the IDS uniformly with respect to the energy variable has been pursued, see for instance [49] and the references therein. Independently of the approximability by finite volume eigenvalue counting functions it is possible to give an abstract definition of the IDS by an averaged trace per unit volume formula, see [5, 47, 53, 62]. In the amenable setting, both definitions of the IDS coincide. For a certain class of metric graphs the approximability of the IDS has been established before. In [19, 20, 27] this has been carried out for random metric graphs with an Zd structure. A step of the proof which is specific to the setting of quantum graphs concerns the influence of finite rank perturbations on eigenvalue counting functions. When one considers Laplacians on manifolds, one would rather use the principle of not feeling the boundary of heat kernels, cf. e.g. [2, 52, 61], to derive the analogous step of the proof. Next we discuss the literature on Wegner estimates and on the regularity of the IDS. Wegner gave in [69] convincing arguments for the Lipschitz continuity of the IDS of the discrete Anderson model on 2 (Zd ). The proof is based on an estimate for the expected number of eigenvalues in a finite energy interval of a restricted box Hamiltonian. A rigorous proof of the latter estimate was

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given in [29] (for the analogous alloy-type model on L2 (Rd )). However, the bound of [29] was not sufficient to establish the Lipschitz continuity of the IDS. In [12] tools to prove Hölder continuity were supplied, see also [24]. They concern bounds on the spectral shift function. Up to now the most widely applicable result concerning the Lipschitz-continuity of the IDS is given in [11]. An alternative approach to derive Lipschitz continuity of the IDS goes via spectral averaging of resolvents, see [10, 40]. However, this method requires more assumptions on the underlying model. Wegner’s estimate and all references mentioned so far concern the case where the random variables couple to a perturbation which is a non-negative operator. If this is not the case, additional ideas are necessary to obtain the desired bounds, see [23, 33, 46, 65, 67]. In our situation, where the perturbation concerns the metric of the underlying space, the dependence on the random variables is not monotone. This is also the case for random metrics on manifolds studied in [51]. To deal with non-monotonicity, the proof of the Wegner estimate (Theorem 2.9) takes up an idea developed in [51], which is not unrelated to [33]. The relevant formula used in the proof is (5.2). We need also a partial integration formula whose usefulness was first seen in [23]. In the context of quantum graphs it is not necessary to rely on sophisticated estimates on the spectral shift function. It is sufficient to adapt a finite rank perturbation bound, which was used in [45] for the analysis of one-dimensional random Schrödinger operators. These estimates are closely related to the finite rank estimates mentioned earlier in the context of the approximability of the IDS. For Schrödinger operators on metric graphs where the randomness enters via the potential, Wegner estimates have been proved in [18, 21, 27]. In the recent preprint [39] a Wegner estimate for a model with Zd -structure and random edge lengths has been established. The proof is based on different methods than we use in the present paper. Next, we want to explain an application of Wegner estimates apart from the continuity of the IDS. It concerns the phenomenon of localisation of waves in random media. More precisely, for certain types of random Schrödinger operators on 2 (Zd ) and on L2 (Rd ) it is well known that in certain energy intervals near spectral boundaries the spectrum is pure point. There are two basic methods to establish this fact (apart form the one-dimensional situation where specific methods apply). The first one is called multiscale analysis and was invented in [17]. The second approach from [1] is called fractional moment method or Aizenman-Molchanov method. A certain step of the localisation proof via multiscale analysis concerns the control of spectral resonances of finite box Hamiltonians. A possibility to achieve this control is the use of a Wegner estimate. In fact, the Wegner estimates needed for this purpose are much weaker than those necessary to establish regularity of the IDS. This has been discussed in the context of random quantum graphs in Section 3.2 of [18]. Recently localisation has been proven for several types of random quantum graphs. In [14, 38, 39] this has been done for models with Zd -structure, while [26] considers operators on tree-graphs. On the other hand, delocalisation, i.e. existence of absolutely continuous spectrum, for quantum graph

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models on trees has been shown in [3]. This result should be seen in the context of earlier, similar results for combinatorial tree graphs [3, 4, 16, 31, 32]. Now let us discuss some open questions concerning random quantum graph models. As for models on 2 (Zd ), proofs of localisation require that the random variables entering the operators should have a regular distribution. In particular, if the law of the variables is a Bernoulli measure, no known proof of localisation applies. This is different for random Schrödinger operators L2 (Rd ). Using a quantitative version of the unique continuation principle for solutions of Schrödinger equations, localisation was established in [6] for certain models with Bernoulli disorder. The proof does not carry over to the analogous model on 2 (Zd ), since there is no appropriate version of the unique continuation principle available. For random quantum graphs the situation is even worse, since they exhibit in great generality compactly supported eigenfunctions, even if the underlying graph is Zd . Like for random, ergodic Schrödinger operators on 2 (Zd ) and on L2 (Rd ) there is no proof of delocalisation for random quantum graphs with Zd structure. In the above mentioned papers on delocalisation it was essential that the underlying graph is a tree. An even harder question concerns the mobility edge. Based on physical reasoning one expects that localised point spectrum and delocalised absolutely continuous spectrum should be separated in disjoint intervals by mobility edges. In the context of random operators where the disorder enters via the geometry this leads to an intriguing question pointed out already in [9]. If one considers a graph over Zd which is diluted by a percolation process, the Laplacian on the resulting combinatorial or metric graph has a discontinuous IDS. In fact, the set of jumps can be characterised rather explicitly and is dense in the spectrum [9, 20, 66]. Now the question is, where the eigenvalues of these strongly localised states repel in some manner absolutely continuous spectrum (if it exists at all). For the interested reader we provide here references to textbook accounts of the issues discussed above. They concern the more classical models on L2 (Rd ) and 2 (Zd ), rather than quantum graphs. In [68] one can find a detailed discussion and proofs of the approximability of the IDS by its finite volume analogues and of Wegner estimates. The survey article [36] is devoted to the IDS in general, while the multiscale proof of localisation is exposed in the monograph [63]. The theory of random Schrödinger operators is presented from a broader perspective in the books [13, 59] and in the summer school notes [28, 30].

2 Basic Notions, Model and Results In the following subsections, we fix basic notions (metric graphs, Laplacians and Schrödinger operators with vertex conditions), introduce the random length model and state our main results. For general treatments and further references on metric graphs, we refer to [15].

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2.1 Metric Graphs Since our random model concerns a perturbation of the metric structure of a graph, we carefully distinguish between combinatorial, topological and metric graphs. A combinatorial graph G = (V, E, ∂) is given by a countable vertex set V, a countable set E of edge labels and a map ∂(e) = {v1 , v2 } from the edge labels to (unordered) pairs of vertices. If v1 = v2 , we call e a loop. Note that this definition allows multiple edges, but we only consider locally finite combinatorial graphs, i.e., every vertex has only finitely many adjacent edges. A topological graph X is a topological model of a combinatorial graph together with a choice of directions on the edges: Definition 2.1 A (directed) topological graph is a CW-complex X containing only (countably many) 0- and 1-cells. The set V = V(X) ⊂ X of 0-cells is called the set of vertices. The 1-cells of X are called (topological) edges and are labeled by the elements of E = E(X) (the (combinatorial) edges), i.e., for every edge e ∈ E, there is a continuous map e : [0, 1] −→ X whose image is the corresponding (closed) 1-cell, and e : (0, 1) −→ e ((0, 1)) ⊂ X is a homeomorphism. A 1-cell is called a loop if e (0) = e (1). The map ∂ = (∂− , ∂+ ) : E −→ V × V describes the direction of the edges and is defined by ∂− e := e (0) ∈ V,

∂+ e := e (1) ∈ V.

For v ∈ V we define Ev± = Ev± (X) := { e ∈ E | ∂± e = v } . The set of all adjacent edges is defined as the disjoint union1 Ev = Ev (X) := Ev+ (X) ∪· Ev− (X). The degree of a vertex v ∈ V in X is defined as       deg v = deg X (v) :=  Ev  =  Ev+  +  Ev− . A topological subgraph  is a CW-subcomplex of X, and therefore  is itself a topological graph with (possible empty) boundary ∂ :=  ∩ c ⊂ V(X). Since a topological graph is a topological space, we can introduce the space C(X) of C-valued continuous functions and the associated notion of measurability. A metric graph is a topological graph where we assign a length to every edge. Definition 2.2 A (directed) metric graph (X, ) is a topological graph X together with a length function  : E(X) −→ (0, ∞). The length function induces an identification of the interval Ie := [0, (e)] with the edge e ([0, 1]) (up to

1 The

disjoint union is necessary in order to obtain two different labels in Ev (X) for a loop.

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the end-points of the corresponding 1-cell, which may be identified in X if e is a loop) via the map   x e : Ie −→ X, e (x) = e . (e) Note that every topological graph X can be canonically regarded as a metric graph where all edges have length one. The corresponding length function ½ E(X) is denoted by 0 . In our random model, we will consider a fixed topological graph X with a random perturbation ω of this length function 0 . To simplify matters, we canonically identify a metric graph (X, ) with the disjoint union X of the intervals Ie for all e ∈ E subject to appropriate identifications of the end-points of these intervals (according to the combinatorial structure of the graph), namely  · Ie /∼. X := e∈E

The coordinate maps {e }e can be glued together to a map  : X −→ X.

(2.1)

Remark 2.3 A metric graph is canonnically equipped with a metric and a measure. Given the information about the length of edges, each path in X has a well defined length. The distance between two arbitrary points x, y ∈ X is defined as the infimum of the lengths of paths joining the two points. The measure on X is defined in the following way. For each measurable  ⊂ X the sets  ∩ ψe (Ie ) are measurable as well, and are assigned the Lebesgue measure of the preimage ψe−1 ( ∩ ψe (Ie )). Consequently, we define the volume of  by     vol(, ) := λ ψe−1  ∩ ψe (Ie ) (2.2) e∈E

Using the identification (2.1), we define the function space L2 (X, ) as L2 (Ie ), f = { fe }e with fe ∈ L2 (Ie ) and L2 (X, ) := e∈E

f 2L2 (X,) =



e∈E

| fe (x)|2 dx. Ie

2.2 Operators and Vertex Conditions For a given metric graph (X, ), we introduce the operator dfe (x), dx where the derivative is taken in the interval Ie = [0, (e)]. Note that both the norm in L2 (X, ) and D = D depend on the length function. This observation (Df )e (x) = (D f )e (x) =

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is particularly important in our random length model below, where we perturb the canonical length function 0 = ½ E(X) and therefore have (a priori) different spaces on which a function f lives. Our point of view is that f is a function on the fixed underlying topological graph X, and that the metric spaces are canonically identified via the maps −1 ◦  : (X, ) −→ (X, 0 ). One easily 0 checks that

 f 2L2 (X,) = (e) | fe (x)|2 dx, (2.3a) e∈E

(D f )e (x) =

(0,1)

1 (D0 f )e (e)



 1 x , (e)

(2.3b)

where fe and D0 f on the right side are considered as functions on [0, 1] via ◦  . the identification −1 0 Next we introduce general vertex conditions for Laplacians (X,) = −D 2 and Schrödinger operators H(X,) = (X,) + q with real-valued potentials q ∈ L∞ (X). The maximal or decoupled Sobolev space of order k on (X, ) is defined by Hkmax (X, ) := Hk (Ie ) e∈E

f 2Hk

max (X,)

:=



fe 2Hk (Ie ) .

e∈E k Note that D : Hk+1 max (X, ) −→ Hmax (X, ) is a bounded operator.  We introduce the following two different evaluation maps H1max (X, ) −→ v∈V C Ev : fe (0), − fe (0), if v = ∂− e, if v = ∂− e, and f (v) := f e (v) := − →e fe ((e)), if v = ∂+ e, fe ((e)), if v = ∂+ e,

and f (v) = { f e (v)}e∈Ev ∈ C Ev , f (v) = { f (v)}e∈Ev ∈ C Ev . It follows from stan− → − →e dard Sobolev estimates (see e.g. [43, Lemma 8]) that these evaluation maps are bounded by max{(2/min )1/2 , 1}, provided the minimal edge length 0 < min := inf (e) e∈E

(2.4)

is strictly positive. The second evaluation map is used in connection with the derivative Df of a function f ∈ H2max (X, ). Note that Df is independent of the −→ orientation of the edge. A single-vertex condition at v ∈ V is given by a Lagrangian subspace L(v) of the Hermitian symplectic vector space (C Ev ⊕ C Ev , ηv ) with canonical twoform ηv defined by   ηv (x, x ), (y, y ) := x , y − x, y ,

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where ·, · denotes the standard unitary inner product in C Ev . The set of all Lagrangian subspaces of (C Ev ⊕ C Ev , ηv ) is denoted by Lv and has a natural manifold structure (see, e.g., [22, 41] for more details on these notions). A Lagrangian subspace L(v) can uniquely be described by the pair (Q(v), R(v)) where Q(v) is an orthogonal projection in C Ev with range G (v) := ran Q(v) and R(v) is a symmetric operator on G (v) such that 

 L(v) := (x, x )  (1 − Q(v))x = 0, Q(v)x = R(v)x (2.5) (see e.g. [43]). A field of single-vertex conditions L := {L(v)}v∈V is called a vertex condition. We say that L is bounded, if C R := sup R(v) < ∞, v∈V

(2.6)

where the norm is the operator norm on G (v). For any such bounded vertex condition L, a bounded potential q and a metric graph (X, ) with min > 0, we obtain a self-adjoint Schrödinger operator H(X,),L = (X,),L + q, by choosing the domain

 dom H(X,),L := f ∈ H2max (X, )|( f (v), Df (v)) ∈ L(v) for all v ∈ V . −→ Of particular interest are the following vertex conditions with vanishing vertex operator R(v) = 0 for all v ∈ V: Dirichlet vertex conditions (where L(v) = {0} ⊕ C Ev or G (v) = {0}), Kirchhoff (also known as free) vertex conditions (where (x, x ) ∈ L(v) if all components of x are equal and the sum of all components of x add up to zero, or equivalently G (v) = C(1, . . . , 1)) and Neumann vertex conditions (where L(v) = C Ev ⊕ {0} or equivalently G (v) = C Ev ). 2.3 Random Length Model The underlying geometric structure of a random length model is a random length metric graph. A random length metric graph is based on a fixed topological graph X with V and E the sets of vertices and edges of X, a probability space (, P), and a measurable map  :  × E −→ (0, ∞), which describes the random dependence of the edge lengths. We also assume that there are ω-independent constants min , max > 0 such that min  ω (e)  max for all ω ∈  and e ∈ E. We will use the notation ω (e) := (ω, e). A random length model associates to such a geometric structure (X, , P, ) a random family of Schrödinger operators Hω , by additionally introducing measurable maps L(v) :  −→ Lv for all v ∈ V, and q :  × X −→ R, describing the random dependence of the vertex conditions and the potentials of these operators. We will use the notation Lω := {Lω (v)}v∈V and qω (x) = q(ω, x). We assume that we have constants C R , Cpot > 0 such that qω ∞  Cpot

and

Rω (v)  C R

(2.7)

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for almost all ω ∈  and all v ∈ V, where Rω (v) is the vertex operator associated to Lω (v). From (2.7) and the lower length bound (2.4) it follows that the Schrödinger operators Hω := ω + qω are self-adjoint and bounded from below by some constant λ0 ∈ R uniformly in ω ∈  (see Lemma 4.1). We call the tuple (X, , P, , L, q) a random length model with associated Laplacians and Schrödinger operators ω and Hω and underlying random metric graphs (X, ω ). 2.4 Approximation of the IDS Via Exhaustions Let us describe the setting, for which our first main result holds. Assumption 2.4 Let (X, , P, , L, q) be a random length model with the following properties: (i) The topological graph X is non-compact and connected with underlying (undirected) combinatorial graph G = (V, E, ∂). There is a subgroup ⊂ Aut(G), acting freely on V with only finitely many orbits. Then acts also canonically on X (but does not necessarily respect the directions) by γ e (x) if ∂± (γ e) = γ (∂± e), γ e (x) = γ e (1 − x) if ∂± (γ e) = γ (∂∓ e). This action carries over to -actions on the metric graphs (X, 0 ) and (X, ω ) via the identification (2.1). Note that acts even isometrically on the equilateral graph (X, 0 ) with 0 = ½ E . We can think of (X, 0 ) as a covering of the compact topological graph (X/ , 0 ). (ii) We also assume that acts ergodically on (, P) by measure preserving transformations with the following consistencies between the two

-actions on X and : Metric consistency:

We assume that γ ω (e) = ω (γ e)

(2.8a)

for all γ ∈ , ω ∈  and e ∈ E. This implies that for every γ ∈ , the map γ : (X, ω ) −→ (X, γ ω ) is an isometry between two (different) metric graphs. Moreover, the induced operators U (ω,γ ) : L2 (X, γ −1 ω) ) −→ L2 (X, ω ) Operator consistency:

are unitary. The transformation behaviour of qω and Lω is such that we have for all ω ∈ , γ ∈ , ∗ Hω = U (ω.γ ) Hγ −1 ω U (ω,γ ).

(2.8b)

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Such a random length model (X, , P, , L, q) is called a random length covering model with associated operators Hω and covering group . Remark 2.5 The simplest random length covering model is given when the probability space  consists of only one element with probability 1. In this case, we have only one length function  = ω , one vertex condition L = Lω , and one potential q = qω . The corresponding family of operators consists then of a single operator H = Hω . Moreover, the metric consistency means that acts isometrically on (X, ), and the operator consistency is nothing but the periodicity of H, i.e., the property that H commutes with the induced unitary

-action on L2 (X, ). Next, we introduce some more notation. Let F0 be a relatively compact topological fundamental domain of the -action on (X, 0 ) such that its closure F = F 0 is a topological subgraph. (An example of such a topological fundamental domain is given in Fig. 2a.) There is a canonical spectral distribution function N(λ), associated to the family Hω , given by the trace formula   1 N(λ) := E tr• [½F P• ((−∞, λ])] , (2.9) E(vol(F , • )) where E(·) denotes the expectation in (, P), trω is the trace on the Hilbert space L2 (X, ω ), and Pω (I) denotes the spectral projection associated to Hω and the interval I ⊂ R. Moreover, the volume vol(F , • ) is defined in (2.2). The function N is called the (abstract) integrated density of states with abbreviation IDS. In the case of an amenable group the abstract IDS can also be obtained via appropriate exhaustions. This is the statement of Theorem 2.6 below. A discrete group is called amenable, if there exist a sequence In ⊂ of finite, non-empty subsets with lim

n→∞

|In In γ | = 0, |In |

for all

γ ∈ .

(2.10)

A sequence In satisfying (2.10) is called a Følner sequence.  For every non-empty finite subset I ⊂ , we define (I) := γ ∈I γ F . A sequence In ⊂ of finite subsets is Følner if and only if the associated sequence n = (In ) of topological subgraphs satisfies the van Hove condition lim

n→∞

|∂(In )| = 0. vol((In ), 0 )

(2.11)

The proof of this fact is analogous to the proof of [61, Lemma 2.4] in the Riemannian manifold case. Note that (2.11) still holds if we replace ∂(In ) by ∂r (In ) for any r  1, where ∂r  denotes the thickened combinatorial boundary { v ∈ V | d(v, ∂)  r } and d denotes the combinatorial distance which agrees (on the set of vertices) with the distance function of the unilateral metric graph (X, 0 ).

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A Følner sequence In is called tempered, if we additionally have  | kn In+1 Ik−1 | sup < ∞. |In+1 | n∈N

(2.12)

Tempered Følner sequences are needed for an ergodic theorem of Lindenstrauss [48]. This ergodic theorem plays a crucial role in the proof of Theorem 2.6 presented below. However, the additional property (2.12) is not very restrictive since it was also shown in [48] that every Følner sequence In has a tempered subsequence In j . For any compact topological subgraph  of X, we denote the operator with Dirichlet vertex conditions on the boundary vertices ∂ and with the original vertex conditions Lω (v) on all inner vertices v ∈ V() \ ∂ by Hω,D . The label D refers to the Dirichlet conditions on ∂. For a precise definition of the Dirichlet operator via quadratic forms, we refer to Section 4. The spectral projection corresponding to Hω,D is denoted by Pω,D . It is well-known that compactness of  implies that the operator Hω,D has purely discrete spectrum. The normalised eigenvalue counting function associated to the operator Hω,D is defined as   1 Nω (λ) = trω Pω,D ((−∞, λ]) . vol(, ω ) The function Nω is the distribution function of a (unique) pure point measure which we denote by μ ω. If  = (In ) is associated to a Følner sequence In ⊂ , we use the abbreviations Hωn,D := Hω(In ),D for the Schrödinger operator with Dirichlet conditions on ∂(In ), Nωn := Nω(In ) for the normalised eigenvalue counting function and n) μnω := μ(I for the corresponding pure point measure on (In ). We can now ω state our first main result: Theorem 2.6 Let (X, , P, , L, q) be a random length covering model as described in Assumption 2.4 with amenable covering group . Let N be the IDS of the operator family Hω . Then there exist a subset 0 ⊂  of full P-measure such that we have, for every tempered Følner sequence In ⊂ , lim Nωn (λ) = N(λ)

n→∞

for all ω ∈ 0 and all points λ ∈ R at which N is continuous. The proof is given in Section 4. Remark 2.7 The proof of Theorem 2.6 yields even more. Let μ denote the measure associated to the distribution function N. Then we have lim μnω ( f ) = μ( f )

j→∞

(2.13)

for all ω ∈ 0 and all functions f of the form f (x) = g(x)(x + 1)−1 with a function g continuous on [0, ∞) and with limit at infinity. (The behaviour of g(x)

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for x < 0 is of no importance since the spectral measures of all operators under consideration are supported on R+ = [0, ∞).) 2.5 Wegner Estimate In this subsection, we state a linear Wegner estimate for Laplace operators of a random length model with independently distributed edge lengths and fixed Kirchhoff vertex conditions. This Wegner estimate is linear both in the number of edges and in the length of the considered energy interval. As mentioned in the introduction, a similar result for the case Zd was proved recently by different methods in [39]. In contrast to the previous subsection, we do not require periodicity of the graph X associated to a group action. More precisely, we assume the following: Assumption 2.8 Let (X, , P, , L, q) be a random length model with the following properties: (i) We have q ≡ 0, i.e., the random family of operators are just the Laplacians (Hω = ω ) and we have no randomness in the vertex condition by fixing L to be Kirchhoff in all vertices. Thus it suffices to look at the tuple (X, , P, ). (ii) We have a uniform upper bound dmax < ∞ on the vertex degrees deg v, v ∈ V(X). (iii) Since the only randomness occurs in the edge lengths satisfying 0 < min  ω (e)  max

for all ω ∈  and e ∈ E(X),  we think of the probability space  as a Cartesian product e∈E [min , max ] with projections   ω → ωe = ω (e) ∈ [min , max ]. The measure P is assumed to be a product e∈E Pe of probability measures Pe . Moreover, for every e ∈ E, we assume that Pe is absolutely continuous with respect to the Lebesgue measure on [min , max ] with density functions he ∈ C1 (R) satisfying he ∞ , h e ∞  Ch ,

(2.14)

for a constant Ch > 0 independent of e ∈ E. Recall that trω is the trace in the Hilbert space L2 (, ω ). In the next theorem on (, ω ) with Pω,D denotes the spectral projection of the Laplacian ,D ω Kirchhoff vertex conditions on all interior vertices and Dirichlet boundary conditions on ∂. Under these assumptions we have: Theorem 2.9 Let (X, , P, ) be a random length model satisfying Assumption 2.8. Let u > 1 and Ju = [1/u, u]. Then there exists a constant C > 0 such that   E tr P•,D (I)  C · λ(I) · |E()|

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for all compact subgraphs  ⊂ X and all compact intervals I ⊂ Ju , where λ(I) denotes the Lebesgue-measure of I, and where |E()| denotes the number of edges in . The constant C > 0 depends only the constants u, dmax , min , max and the bound Ch > 0 associated to the densities he (see (2.14)). The proof will be given in Section 5. We finish this section with the following corollary. Recall that the periodic situation is a special case of a random length covering model (see Remark 2.5): Corollary 2.10 Let (X, , P, ) be a random length covering model, satisfying both Assumptions 2.4 and 2.8, with amenable covering group . Then the IDS N of the Laplacians ω is a continuous function on R and even Lipschitz continuous on (0, ∞). Proof The Lipschitz continuity of N on (0, ∞) follows immediately from Theorems 2.6 and 2.9. It remains to prove continuity of N on (−∞, 0]. Note that our model is a special situation of the general ergodic groupoid setting given in [53]. Thus, N is the distribution function of a spectral measure of the ⊕ direct integral operator  ω dP(ω). Since ω  0 for all ω, N(λ) vanishes for all λ < 0. Moreover, if N would have a jump at λ = 0, then ker ω would be non-trivial for almost all ω ∈ . But ω f = 0 implies 2

   df  (x) dx 0 = f, ω f  =  dx  X since ω has Kirchhoff vertex conditions. Thus f is a constant function. Now X is connected as well as non-compact, which implies that vol(X, ω ) = ∞ by the lower bound min on the lengths of the edges. Hence constant functions are not in L2 . This gives a contradiction.   Our result on Lipschitz continuity of N on (0, ∞) is optimal in the following sense: Remark 2.11 It is well-known that the IDS of the free Laplacian R on R is proportional to the square root of the energy. Note that this does not change when adding Kirchhoff boundary conditions at arbitrary points. Therefore, every model satisfying Assumptions 2.4 and 2.8 for a metric graph isometric to R has in fact the above IDS. Therefore, we cannot expect Lipschitz continuity of the IDS at zero for random length models without further assumptions.

3 Kagome Lattice as an Example of a Planar Graph In this section, we illustrate the concepts of the previous section for an explicit example. We introduce a particular regular tessellation of the Euclidean plane admitting finitely supported eigenfunctions of the combinatorial Laplacian.

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Fig. 1 Illustration of the planar graph X (Kagome lattice)

2

1

We discuss in detail the discontinuities of the IDS of the combinatorial Laplacian and of the Kirchhoff Laplacian of the induced equilateral metric graph. On the other hand, applying Corollary 2.10, we see that the IDS of a random family of Kirchhoff Laplacians for independent distributed edge lengths is continuous. Thus, randomness leads to an improvement of the regularity of the IDS in this example. We consider the infinite planar topological graph X ⊂ C as illustrated in Fig. 1. This graph is sometimes called Kagome lattice. Every vertex of X has degree four and belongs to a uniquely determined upside triangle. Introducing w1 = 1 and w2 = eπ i/3 , we can identify the lower left vertex of a particular upside triangle with the origin in C and its other two vertices with w1 , w2 ∈ C. Consequently, the vertex set of X is given explicitly as the disjoint union of the following three sets:       V(X) = 2Zw1 + 2Zw2 ∪· w1 + 2Zw1 + 2Zw2 ∪· w2 + 2Zw1 + 2Zw2 . A pair v1 , v2 ∈ V = V(X) of vertices is connected by a straight edge if and only if |v2 − v1 | = 1. We write v1 ∼ v2 for adjacent vertices. The above realisation of the planar graph X ⊂ C is an isometric embedding of the metric graph (X, 0 ). The group Z2 acts on X via the maps Tγ (x) := 2γ1 w1 + 2γ2 w2 + x. A topological fundamental domain F0 of X is thickened in Fig. 2a. The set of

2

2 5

2

b

1

c

2

b a b

1

4

a c

3

H γ0

1 2

b

(a)

(b)

Fig. 2 a The periodic graph with thickened topological fundamental domain F0 and combinatorial fundamental domain Q = {a, b , c} b If γ0 is vertically extremal for F, all white encircled vertices are zeroes of F

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vertices of the topological subgraph F = F0 (obtained by taking the closure of F0 considered as a subset of the metric space (X, )) is given by {a, b , c, a, b , b , c }. Note that we have to distinguish carefully between a topological and a combinatorial fundamental domain. Let G denote the underlying combinatorial graph with set V of vertices and E of combinatorial edges. The maps Tγ act also on the set of vertices V and a combinatorial fundamental domain is given by Q = {a, b , c}. We denote the translates Tγ (Q) of Q by Qγ . 3.1 Spectrum and IDS of the Combinatorial Laplacian We first observe that G admits finitely supported eigenfunctions of the combinatorial Laplacian comb : Choose an arbitrary hexagon H ⊂ X with vertices {u0 , u1 , . . . , u5 }. Then there exists a centre w0 ∈ C of H such that we have

 {u0 , u1 , . . . , u5 } = w0 + ekπ i/3 | k = 0, 1, . . . , 5 . The following function F H : V −→ {0, ±1} on the vertices 0, if v ∈ V \ {u0 , . . . , u5 }, F H (v) := k (−1) , if v = w0 + ekπ i/3 ,

(3.1)

satisfies comb F H (v) =

1  3 (F H (v) − F H (w)) = F H (v). deg(v) w∼v 2

Thus, the vertices of every hexagon H ⊂ X are the support of a combinatorial eigenfunction F H : V −→ R. The functions F H are the only finitely supported eigenfunctions up to linear combinations: Proposition 3.1 (a) Let F : V −→ R be a combinatorial eigenfunction on X with finite support supp F ⊂ V. Then 3 comb F = F 2 and F is a linear combination of finitely many eigenfunctions F H of the above type (3.1). (b) Let Hi (i = 1, . . . , k) be a collection of distinct, albeit not necessarily disjoint, hexagons, and Fi := F Hi the associated compactly supported eigenfunctions. Then the set F1 , . . . , Fk is linearly independent. (c) If g ∈ 2 (V) satisfies comb g = μg, then μ = 3/2. (d) The space of 2 (V)-eigenfunctions to the eigenvalue 3/2 is spanned by compactly supported eigenfunctions.

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Proof To prove (a), assume that F : V −→ R is a finitely supported eigenfunction. Let Q = {a, b , c} be a combinatorial fundamental domain of Z2 , as illustrated in Fig. 2a and Qγ := Tγ (Q). Let Hγ be the uniquely defined hexagon containing the three vertices Qγ . Moreover, we define

 A0 := γ ∈ Z2 | supp F ∩ Qγ = ∅ . Let ε1 = (1, 0) and ε2 = (0, 1). We say that γ0 = (γ01 , γ02 ) ∈ A0 is vertically extremal for F, if the second coordinate γ02 is maximal amongst all γ ∈ A0 and if γ0 − ε1 ∈ / A0 . This means that F vanishes in the left neighbour of Qγ0 and in all vertices vertically above Qγ0 . Hence, γ0 in Fig. 2b is vertically extremal if F vanishes in all white encircled vertices and does not vanish in at least one of the black vertices. Obviously, A0 has always vertically extremal elements. Choosing such a γ0 ∈ A0 , we will show below that F is an eigenfunction with eigenvalue 3/2 and that the following facts hold: (i) γ0 + ε1 belongs to A0 , (ii) γ0 − ε2 or γ0 − ε2 − ε1 belong to A0 , (iii) adding a suitable multiple of F Hγ0 to F, we obtain a new eigenfunction

 F1 and a set A1 := γ ∈ Z2 | supp F1 ∩ Qγ = ∅ satisfying γ0 ∈ / A1 ,



A1 \ A0 ⊂ γ0 − ε2 , γ0 + ε1 − ε2 .

To see this, let γ0 ∈ A0 be vertically extremal and v1 , . . . , v5 , w1 , w2 be chosen as in Fig. 2b. The eigenvalue equation at the vertices v4 and v5 , in which F vanishes, imply that we have F(v1 ) = −F(v2 ) = F(v3 ) = 0. Applying the eigenvalue equation again, now at v2 , yields that the eigenvalue of F must be 3/2. If γ0 + ε1 ∈ / A0 , F would vanish in w1 and all its neighbours, except for v3 . This would contradict to the eigenvalue equation at w1 and (i) is proven. Similarly, if γ0 − ε2 , γ0 − ε2 − ε1 ∈ / A0 , we would obtain a contradiction to the eigenvalue equation at the vertex w2 . This proves (ii). By adding F(v1 )F Hγ0 to F, we obtain a new eigenfunction F1 (again to the eigenvalue 3/2) which vanishes at all vertices of Qγ0 = {v1 , v2 , v3 }. Thus we / A1 . But F and F1 differ only in the vertices Qγ0 , Qγ0 +ε1 , Qγ0 −ε2 and have γ0 ∈ Qγ0 +ε1 −ε2 , establishing property (iii). The above procedure can be iteratively (from left to right) applied to the hexagons in the top row of A0 : Step (iii) can be applied to the function F1 and a vertically extremal element of A1 . After a finite number n of steps the top row of hexagons in A0 is no longer in the support of the function Fn . (Note that property (i) implies that when removing the penultimate hexagon form the right, one has simultaneously removed the rightermost one, too.) Again, this procedure can be iterated removing successively rows of hexagons. This

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time property (ii) guarantees that the procedure stops after a finite number N of steps with F N ≡ 0. We have proven statement (a). Now we turn to the proof of (b). Since the graph is connected there exists a k Hi which is adjacent to some vertex outside A. Then v is vertex v in A := ∪i=1 contained in precisely one hexagon Hi0 . (In the full graph each vertex is in two hexagons.) Thus the condition k 

αi Fi = 0

αi ∈ C

(3.2)

i=1

evaluated at the vertex v implies αi0 = 0. This shows that all coefficients αi in (3.2) corresponding to hexagons Hi lying at the boundary of A vanish. This leads to an equation analogous to (3.2) where the indices in the sum run over a strict subset of {1, . . . , k}. Now one iterates the procedure and shows that actually all coefficients α1 , . . . , αk in (3.2) are zero. We have shown linear independence of F1 , . . . , Fk . To prove (c) we recall that the IDS comb is a spectral measure (see e.g. [53, Proposition 5.2]). Thus the IDS jumps at the value μ. This in turn implies by [66, Proposition 5.2] that there is a compactly supported g˜ satisfying the eigenvalue equation. Now (a) implies μ = 3/2. Statement (d) follows from [54, Theorem 2.2], cf. also the proof of Proposition 3.3.   We are primarily interested in 2 -eigenfunctions of comb , since their eigenvalues coincide with the discontinuities of the corresponding IDS. For combinatorial covering graphs with amenable covering group , every 2 eigenfunction F implies the existence of a finitely supported eigenfunction to the same eigenvalue which is implied, e.g., by [66, Proposition 5.2] or [54, Theorem 2.2]. (Related, but different results have been obtained before in [57]. If the group is even abelian, as is the case for the Kagome lattice, the analogous result was proven even earlier in [42].) It should be mentioned here that the situation is very different in the smooth category of Riemannian manifolds. There, compactly supported eigenfunctions cannot occur due to the unique continuation principle. In the discrete setting of graphs, non-existence of finitely supported combinatorial eigenfunctions is—at present—only be proved for particular examples or in the case of planar graphs of non-positive combinatorial curvature; see [34] for more details. Hence, Proposition 3.1 tells us that X does not admit combinatorial 2 -eigenfunctions associated to eigenvalues μ = 3/2. Next, let us discuss spectral informations which can be obtained with the help of Floquet theory. Using a general result of Kuchment (see [42] or [44, Theorem 8]) for periodic finite difference operators (applying Floquet theory to such operators) we conclude that the compactly supported eigenfunctions of comb associated to the eigenvalue 3/2 are already dense in the

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whole eigenspace ker( comb − 3/2). As for the whole spectrum, we derive the following result:     Proposition 3.2 Denote by σac comb and σp comb the absolutely continuous and point spectrum of comb on our Z2 -periodic graph X. Then we have     3 σac comb = 0, 2

  σp comb =

and

  3 . 2

The proof follows from standard Floquet theory (for a similar hexagonal graph model see [37]): Proof Note that we have the unitary equivalence comb

∼ =



⊕ T2

θcomb dθ,

where θcomb is the θ-equivariant Laplacian on Q, θ ∈ T2 := R2 /(2π Z)2 . This operator is equivalent to the matrix ⎛

−1 − e−iθ2

4

1⎜ θcomb ∼ = ⎝ −1 − eiθ2 4 −eiθ1 − eiθ2

4

−e−iθ1 − e−iθ2

⎞

⎟ −1 − e−iθ1 ⎠

−1 − eiθ1

4

using the basis F ∼ = (F(a), F(b ), F(c)) for a function on Q and the fact that F(Tγ v) = ei θ,γ  F(v) (equivariance). The characteristic polynomial is     3 2 3 + 2κ 3 μ− p(μ) = μ − − 2 4 16

,

where κ = cos θ1 + cos θ2 + cos(θ1 − θ2 ), and the eigenvalues of θcomb are μ1 =

3 2

and μ± =

3 1√ ± 3 + 2κ. 4 4

In particular, we recover the fact that comb has an eigenfunction, since μ1 is independent of θ, only μ± depend on θ via κ = κ(θ ). Note that we have 3 − =κ 2



2π 4π , 3 3

  κ(θ)  κ(0, 0) = 3,

giving the spectral bands B− = [0, 3/4] and B+ = [3/4, 3/2].

 

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The next result discusses (dis)continuity properties of the IDS associated to the combinatorial Laplacian on X: Proposition 3.3 Let Ncomb be the (abstract) IDS of the Z2 -periodic operator comb , given by Ncomb (μ) =

  1 tr ½ Q Pcomb ((−∞, μ]) , |Q|

where tr is the trace on the Hilbert space 2 (V) and Pcomb denotes the spectral projection of comb . Then Ncomb vanishes on (−∞, 0], is continuous on R \ {3/2} and has a jump of size 1/3 at μ = 3/2. Moreover, Ncomb is strictly monotone increasing on [0, 3/2] and Ncomb (μ) = 1 for μ  3/2. Proof The following facts are given, e.g., in [57, p. 119]:   (i) the points of increase of Ncomb coincide with  thespectrum σ comb and (ii) Ncomb can only have discontinuities at σ p comb . Together with Proposition 3.2, all statements of the proposition follow, except for the size of the jump at μ = 3/2.  Let us choose a Følner sequence In ⊂ Z2 and define n = γ ∈In Qγ . Let ∂n denote the set of boundary vertices of the combinatorial graph induced by the vertex set n , and ∂r n := { v ∈ V(X) | d(v, ∂n )  r }

(3.3)

be the thickened (combinatorial) boundary. Let D(μ) := Ncomb (μ) − lim Ncomb (μ − ε) = ε→0

  1 tr ½n Pcomb ({μ}) . |n |

(3.4)

The last equality in (3.4) holds for all n and follows easily from the Z2 invariance of the operator comb . It remains to prove that D(3/2) = 1/3. Let  n = n \ ∂1 n and Dn (μ) :=

1 dim En (μ), |n |

 where En (μ) := F ∈ ker( comb − μ) | supp F ⊂  n . Arguments as in [56] or in [54] show that D(μ) = lim Dn (μ). n→∞

(3.5)

For the convenience of the reader, we outline the proof of (3.5) below. Using part (b) of Proposition 3.1 one can show that dim En (μ) equals up to a boundary term the number of hexagons contained in  n . Since every translated

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combinatorial fundamental domain Qγ uniquely determines a hexagon Hγ and |Q| = 3, we conclude that dim En (μ) ≈ 13 |n |, up to an error proportional to |∂1 n |. The van Hove property (2.11) (which holds also in the combinatorial setting) then implies the desired result D(3/2) = limn→∞ Dn (3/2) = 1/3.   Finally, we outline the proof of (3.5): Let E(μ) = ker comb −μ and Sn (μ) = ½n E(μ). Let bn : Sn (μ) −→ R|∂1 n | be the boundary map, i.e., bn (F) is the collection of all values of F assumed at the (thickened) boundary vertices ∂1 n . Then ker bn = En (μ) ⊂ Sn (μ), and we have Dn (μ)  D(μ) 

dim Sn (μ) dim ker bn dim ran bn |∂1 n | = +  Dn (μ) + , |n | |n | |n | |n |

which yields (3.5), by taking the limit, as n → ∞.

 

3.2 Spectrum and IDS of the Periodic Kirchhoff Laplacian   There is a well known correspondence between the spectrum σ comb on a graph G and the spectrum of the (Kirchhoff) Laplacian 0 on the corresponding (equilateral) metric graph (X, 0 ) with 0 = ½ E (see e.g. [7,  8,60, 64] and the references therein). Namely, any λ = k2 π 2 lies in σp 0 resp. √       σac 0 iff μ(λ) = 1 − cos λ lies in σp comb resp. σac comb . Moreover, the eigenspace of the metric Laplacian is isomorphic to the corresponding eigenspace of the combinatorial Laplacian. Let F : V −→ C be a finitely supported eigenfunction of comb as in the previous section. In particular, the eigenvalue must be μ = 3/2. The above mentioned correspondence shows that, for every λ = (2k + 2/3)2 π 2 , k ∈ Z, (i.e. μ(λ) = 3/2)), there is a Kirchhoff eigenfunction f : X −→ R of compact support associated to the eigenvalue λ, satisfying f (v) = F(v) at all vertices v ∈ V. In addition, if λ = k2 π 2 , there are so-called Dirichlet eigenfunctions of 0 , determined by the topology of the graph (see e.g. [44, 50, 64]), which are also generated by compactly supported eigenfunctions. Using the results [7, 8], we conclude from Proposition 3.2: Corollary 3.4 Let 0 denote the Kirchhoff Laplacian of the equilateral metric graph (X, 0 ). Let σp and σac denote the point spectrum and absolutely continuous spectrum and σcomp denote the spectrum given by the compactly supported eigenfunctions. Then we have  !      

 2 2 2  2k + π  k ∈ Z ∪ k2 π 2  k ∈ N σcomp 0 = σp 0 = 3 and 

σac 0



"   # " #      2 2 2 2 2 2 2 2 2 = 0, 2k − π ∪ π , 2k + π . 3 3 3 k∈N

(3.6)

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Similarly, as in the discrete setting, we conclude the following (dis) continuity properties of the IDS: Proposition 3.5 Let N0 be the (abstract) IDS of the Z2 -periodic Kirchhoff Laplacian 0 on the metric graph (X, 0 ), given by N0 (λ) =

  1 tr ½F P0 ((−∞, λ]) , vol(F , 0 )

where tr is the trace on the Hilbert space L2 (X, 0 ) and P0 denotes the spectral projection of 0 . Then all the discontinuities of N0 : R −→ [0, ∞) are (i) at λ = (2k + 23 )2 π 2 , k ∈ Z, with jumps of size 16 , (ii) at λ = k2 π 2 , k ∈ N, with jumps of size 12 . Moreover, N0 is strictly monotone increasing on the absolutely continuous   spectrum σac 0 given in (3.6) and N0 is constant on the complement of σ 0 . Proof Our periodic situation fits into the general setting given in [53], by choosing the trivial probability space  = {ω} with only one element. Proposition 5.2 in [53] states that N0 is the distribution function of a spectral measure for the operator 0 . Consequently, discontinuities of N0 can only occur at the L2 -eigenvalues of 0 , and the points of increase of N0 coincide with the spectrum σ ( 0 ), which is given in Corollary 3.4. Hence, it only remains to prove the statements about the discontinuities of N0 . We know from [44, Theorem 11] that the compactly supported eigenfunctions densely exhaust every L2 -eigenspace of 0 . Let In ⊂ Z2 be a Følner sequence. This time, we look at the corresponding topological graphs (In ) and their thickened topological boundaries ∂r (In ) = { x ∈ X | d(x, ∂(In ))  r }, and denote them by n and ∂r n , respectively. We are interested in the jumps D(λ) := N0 (λ) − lim N0 (λ − ε) = ε→0

  1 tr ½n P0 ({λ}) , vol(n , 0 )

where the right hand side is, again, independent of the choice of n. Let  n be the closure of n \ ∂1 n and Dn (λ) :=

1 dim En (λ), vol(n , 0 )

 with En (λ) = f ∈ ker( 0 − λ) | supp f ⊂  n . Arguments analogously to the proof of (3.5) yield D(λ) = lim Dn (λ). n→∞

(3.7)

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For the proof of (3.7), however, we have to define the boundary map bn : Sn (λ) −→

  C ⊕ C Ev v∈∂n

  by (bn f )v := f (v), Df (v) . −→

Let λ = (2k + 2/3)2 π 2 , k ∈ Z. We follow the same arguments as in the proof of Proposition 3.3. Again, dim En (λ) is equal to the number of hexagons contained in n up to a boundary term and we have vol(F , 0 ) = 6 (see Fig. 2a). Therefore, we derive that the corresponding jump is of size 1/6. Let λ = k2 π 2 , k ∈ N. We know from [64] or from [50, Lemma 5.1 and Proposition 5.2] that the dimension of En (λ) is (up to an error proportional to |∂n |) approximately equal to |E(n )| − |V(n )| ≈

1 vol(n , 0 ). 2

This implies that N0 has a discontinuity at λ = k2 π 2 of size 1/2.

 

Remark 3.6 Note that Propositions 3.3 and 3.5 hold also for general covering graphs X → X0 with amenable covering group and compact quotient X0 ∼ = X/ , once we have information about the shape of the support of elementary eigenfunctions (i.e., eigenfunctions, which generate the eigenspace by linear combinations and translations). In our Kagome lattice example the elementary eigenfunction is supported on a hexagon. For example, the jump of size 1/3 at the eigenvalue μ = 3/2 in the discrete case is the number ν of hexagons determined by a combinatorial fundamental domain (ν = 1) divided by the number of vertices in a combinatorial fundamental domain (|Q| = 3). In the metric graph setting, the jump at λ = (2k + 2/3)2 π 2 is of size 1/6 due to the fact that we have six edges in one topological fundamental domain. For the eigenvalues at λ = k2 π 2 (also called topological, see [50]) we even have a precise information for any r-regular amenable covering graph, namely     2 2 dim En (λ) ≈ |E(n )| − |V(n )| ≈ 1 − |E(n )| = 1 − vol(n , 0 ), r r up to an error proportional to |∂n |, so that the jump of N0 at λ is (1 − 2/r). 3.3 IDS of Associated Random Length Models Finally, we impose a random length structure  :  × E −→ [min , max ] on the edges of (X, 0 ) with independently distributed edge lengths, as described in Assumption 2.8. Then Corollary 2.10 tells us that the associated integrated density of states N : R −→ [0, ∞) is continuous and even Lipschitz continuous on (0, ∞). Hence, all discontinuities occurring for the IDS of the Kirchhoff Laplacian on the Z2 -periodic graph (X, 0 ) disappear by introducing this type of randomness.

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4 Proof of the Approximation of the IDS Via Exhaustions In this section, we prove Theorem 2.6, namely, that the non-random integrated density of states (2.9) can be approximated by suitably chosen normalised eigenvalue counting functions, for P-almost all random parameters ω ∈ . For the following considerations, we need the quadratic forms associated to the Schrödinger operators. Recall that for each Lagrangian subspace Lv ⊂ C Ev ⊕ C Ev describing the vertex condition at v ∈ V there exists a unique orthogonal projection Qv on C Ev with range Gv := ran Qv and a symmetric operator on Gv such that (2.5) holds. Let  ⊂ X be a topological subgraph. The quadratic form associated to the operator with vertex conditions given by (Gv , Rv ) at inner vertices V() \ ∂ and Dirichlet conditions at ∂ is defined as $ %  dom h,D = f ∈ H1max (X, )  f (v) ∈ Gv ∀v ∈ V() \ ∂, f (v) = 0 ∀v ∈ ∂ ,  h,D ( f ) = Df 2L2 (,) + q f , f L2 (,) +

Rv f (v), f (v)Gv . v∈V()

In particular, if  = X is the full graph, then there is no boundary and h = h X is the quadratic form associated to the operator H = H(X,),L . If min := infe (e) > 0, Cpot := q ∞ < ∞ and supv Rv =: C R < ∞, then h,D is a closed quadratic form with corresponding self-adjoint operator H ,D . Lemma 4.1 For any subgraph  of X, the quadratic form h,D is closed. Moreover, the associated self-adjoint operator H ,D has domain given by 

dom H ,D = f ∈ H2max (X, ) f (v) = 0 ∀v ∈ ∂ V,  f (v) ∈ Gv , Qv Df (v) = Rv f (v) ∀v ∈ V() \ ∂ . −→ Moreover, H ,D is uniformly bounded from below by −C0 where C0  0 depends only on − , C R and Cpot , but not on . Proof The first assertion follows from [43, Theorem 17]. The uniform lower bound is a consequence of [43, Corollary 10] where the lower bound is given explicitly. Basically, the statements follow from a standard Sobolev estimate of the type        | f (v)|2  η Df 2 + Cη f 2  Rv f (v), f (v)   C R   v

v∈V()

for η > 0, where Cη depends only on η, C R and min .

 

The Dirichlet operator will serve as upper bound in the bracketing inequality (4.1) later on. In order to have a lower bound we introduce a Neumanntype operator H  via its quadratic form h . Since the vertex conditions can

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be negative, we have to use the boundary condition (C Ev , −C R ) instead of a simple Neumann boundary condition (C Ev , 0). The quadratic form h is defined by $ %  dom h = f ∈ H1max (X, )  f (v) ∈ Gv ∀v ∈ V() \ ∂ ,  h ( f ) = Df 2L2 (,) + q f , f L2 (,) +

Rv f (v), f (v)Gv − CR

 v∈∂

v∈V()\∂

| f (v)|2Gv .

&v = −C R trivially fulfills the norm bound Note that the boundary condition R &v  C R , and therefore by Lemma 4.1, the form h is uniformly bounded R from below by the same constant −C0 as h,D . By adding C0 to the (edge) potential q we may assume that w.l.o.g. H X , H ,D and H  are all non-negative for all subgraphs . We can now show the following bracketing result: Lemma 4.2 Let  be a topological subgraph of X and  be the closure of the complement c . Then



H ,D ⊕ H  ,D  H  H  ⊕ H   0

(4.1)

in the sense of quadratic forms. Proof It is clear from the inclusions {0} ⊂ Gv ⊂ C Ev for all boundary vertices v ∈ ∂ that the quadratic form domains fulfil



dom h,D ⊕ dom h ,D ⊂ dom h ⊂ dom h ⊕ dom h . Moreover, if f = f ⊕ f is in the decoupled Dirichlet domain, then

h,D ( f ) + h ,D ( f ) = h( f )

since f (v) = 0 on boundary vertices, if f ∈ dom h, then

h( f )  h ( f ) + h ( f )

since Rv  −C R . In particular, we have shown the inequality for the quadratic forms.   Next, we provide a useful lemma about the spectral shift function of two operators. For a non-negative operator H with purely discrete spectrum { λk (H) | k  0 } (repeated according to multiplicity), the eigenvalue counting function is given by n(H, λ) := tr ½[0,λ) (H) = |{ k  0 | λk (H)  λ }| .

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The spectral shift function (SSF) of two non-negative operators H1 , H2 with purely discrete spectrum is then defined as ξ(H1 , H2 , λ) := n(H2 , λ) − n(H1 , λ). We have the following estimate: Lemma 4.3 Let (X, , P, ) be a random length metric graph (as described in Subsection 2.3) and  ⊂ X be a compact topological subgraph. Let L1 , L2 be two vertex conditions differing in the vertex set Vdiff ⊂ V() only, and such that the operators (,ω ),Li are non-negative. Let 0  q be a bounded measurable potential and Hi = (,ω ),Li + q. Then we have |ξ(H1 , H2 , λ)|  2



deg v.

(4.2)

v∈Vdiff

Moreover, if ρ : R+ −→ R is a monotone function with ρ ∈ L1 (R+ ), then  |tr[ρ(H1 ) − ρ(H2 )]|  2|ρ(∞) − ρ(0)| deg v, (4.3) v∈Vdiff

where the trace is taken in the Hilbert space L2 (, ω ). Proof Let D0 = dom H1 ∩ dom H2 . Then D0 has finite index in dom Hi , bounded above by twice the number ' of all edges adjacent to vertices v ∈ Vdiff . This implies dim(dom Hi /D0 )  2 v∈Vdiff deg v. Inequality (4.2) follows now from [19, Lemma 9]. The second inequality (4.3) follows readily from Krein’s trace identity

∞ | tr ρ(H1 ) − ρ(H2 )|  |ρ (λ)| · |ξ(H1 , H2 , λ)| dλ. (4.4) 0

  The following uniform resolvent boundedness holds in every random length covering model: Lemma 4.4 Let (X, , P, , L, q) be a random length covering model with covering group , as described in Assumption 2.4, and λ > 0. Then there is a constant Cλ > 0 such that we have tr(Hω + λ)−1  Cλ vol(, 0 ) for all compact subgraphs  ⊂ (X, ) and all ω ∈ . Proof Let Hω,0 denote the  restriction on  with Dirichlet vertex conditions at all vertices. Then Hω,0 = e∈E() Hωe,D , where we identify the edge e with

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the topological subgraph consisting of this edge and its end vertices in X. From (4.2) of Lemma 4.3 we conclude that         tr (H ,0 + λ)−1 − (H  + λ)−1  4  E() = 4 vol(, 0 ). ω ω  λ   λ −1 Since (Hωe,D + λ)−1 is bounded from above by ( e,D ω + λ) , and since the edges are uniformly bounded from above by max , there is a constant cλ > 0 such that tr (Hωe,D + λ)−1  cλ for all e ∈ E() and ω ∈ . This implies the desired estimate with constant Cλ = 4λ−1 + cλ .  

The proof of Theorem 2.6 will now be given in four lemmata. All of these lemmata are based on a given random length covering model (X, , P, , L, q) with an amenable covering group and a fixed tempered Følner sequence In with associated compact topological graphs n := (In ). In the first lemma, we prove the convergence (2.13) for a special family of functions fλ associated to resolvents of the operators. Here, we need to apply an ergodic theorem of Lindenstrauss [48]. In later lemmata we show that the convergence (2.13) carries over to the uniform closure of finite linear combinations of the functions fλ , identify this closure with the help of the Stone-Weierstrass Theorem, and finally conclude the desired convergence for characteristic functions ½[0,λ] at continuity points λ > 0 of the IDS. Lemma 4.5 Let λ > 0 and fλ : [0, ∞) −→ R, fλ (x) = subset 0 ⊂  of full P-measure such that lim

n→∞

1 . x+λ

Then there exists a

     1 1 E tr[½F fλ (H• )] tr fλ Hωn,D = vol(n , ω ) Evol(F , • )

for all ω ∈ 0 . Proof We first consider a fixed ω ∈  and a fixed  = (In ) and suppress the parameters ω and n in the notation. Recall the definitions of H ,D and H  with quadratic form domains given below. Let  denote the closure of the complement c in the metric graph (X, ). By Lemma 4.2 we have (4.1) in the sense of quadratic forms. Since taking inverses is operator monotone, this implies  ,D −1  −1   −1 H ⊕ H  ,D + λ  H+λ  H ⊕ H + λ for all λ > 0. In particular, we obtain inequalities for the following restricted −1 quadratic forms: Set (H + λ)−1  = p (H + λ) i , where i and p denote the canonical inclusions and projections between L2 (, ) and L2 (X, ). Then 

H ,D + λ

−1

  −1  (H + λ)−1 .   H +λ

(4.5)

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,D Consequently, (H + λ)−1 + λ)−1 is non-negative and we have  − (H

 ,D −1   +λ 0  trL2 (,) (H + λ)−1  − H      2  trL2 (,) fλ H  − fλ H ,D  dmax |∂|, λ using Lemma 4.3, where dmax is a finite upper bound on the vertex degree of X, which exists due to the -periodicity of X. Using the van Hove property (2.11) and the estimate min vol(, 0 )  vol(, ω )  max vol(, 0 ), we conclude that lim

n→∞

     n,D  1 tr (Hω + λ)−1 = 0. n − tr fλ Hω vol(n , ω )

(4.6)

Using additivity of the trace and the operator consistency (2.8b), we obtain   trL2 (γ F ,ω ) (Hω + λ)−1 gλ (γ ω), trL2 (n ,ω ) (Hω + λ)−1 n = γF = γ ∈In

where

γ ∈In−1

    gλ (ω) = trL2 (F ,ω ) (Hω + λ)−1 F = tr ½F fλ (Hω ) .

(4.7)

Since, by monotonicity (4.5) and Lemma 4.4, ( −1 ) 0  gλ (ω)  trL2 (F ,ω ) HωF + λ  Cλ vol(F , 0 ), we conclude that gλ ∈ L1 (). Now, we argue as in the proof of Theorem 7 in [52]: Applying Lindenstrauss’ ergodic theorem separately to both expressions 1  gλ (γ ω) |In | −1 γ ∈In

and

1  vol(F , γ ω ), |In | −1 γ ∈In

we conclude that     1 1 E tr[½F fλ (H• )] tr (Hω + λ)−1 =  n n→∞ vol(n , ω ) E(vol(F , • )) lim

(4.8)

for almost all ω ∈ . The lemma follows now immediately from (4.6) and (4.8).  

 Let us denote by L the set of functions x → fλ (x) = (x + λ)−1 | λ > 0 and by A the · ∞ -closure of the linear span of L and the constant function ½ : [0, ∞) −→ R, ½(x) = 1. Note that, by monotonicity (4.5) and Lemma 4.4, both expressions μnω ( f1 ) and μ( f1 ) = (E(vol(F , • )))−1 E(g1 ) (with g1 defined

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in (4.7)) are bounded by a constant K > 0, independent of ω and n. Let 0 ⊂  be the set of full P-measure from Lemma 4.5. Lemma 4.6 Let ω ∈ 0 . Set ν n = f1 · μnω (for n ∈ N) and ν = f1 · μ. Then we have, for all g ∈ A , lim ν n (g) = ν(g).

n→∞

Proof By Lemma 4.5 we know that the statement holds for the function 1 ( fλ − f1 ) for λ = 1. Thus, by linearity and g = ½. We note that fλ · f1 = λ−1 Lemma 4.5, the convergence holds also for all functions g = fλ with λ > 0, λ = 1. To deal with the case λ = 1 note that f1+ε converges to f1 uniformly, as ε → 0. Thus   n ν ( f1 ) − ν n ( f1+ε )  f1 − f1+ε ∞ ν n (½)  Kε. An analogous statement holds for ν n replaced by ν. Thus     ν( f1 ) − ν n ( f1 )  2Kε + ν( f1+ε ) − ν n ( f1+ε ) → 2Kε,

(4.9)

as n → ∞. Since ε > 0 was arbitrary, we conclude that limn→∞ ν n ( f1 ) = ν( f1 ). By linearity, the convergence statement of the Lemma holds for all functions g in the linear span of L ∪ {½}. To show that is holds for all functions in the closure A , as well, one uses uniform approximation and an estimate of the same type as in (4.9).   The next lemma identifies the space A explicitly: Lemma 4.7 The function space A coincides with the set of continuous functions on [0, ∞) which converge at infinity. Proof The statement of the lemma is equivalent to A = C([0, ∞]), where [0, ∞] is the one-point-compactification of [0, ∞). We want to apply the StoneWeierstrass Theorem. Any fλ with λ > 0 separates points and ½ is nowhere vanishing in [0, ∞]. By definition A is a linear space. To show that it is an 1 algebra we use again the formula fλ1 · fλ2 = λ2 −λ ( fλ1 − fλ2 ), which shows that 1 fλ1 · fλ2 ∈ A for λ1 = λ2 . Since A is closed in the sup-norm, we can use an approximation as in the proof of the Lemma 4.6 to show fλ2 ∈ A . A similar argument shows that the product of two limit points f, g of the linear span of L ∪ {½} is in A .   We have established the convergence μnω (g) → μ(g) for all functions of the form g · f1 with g ∈ A . The following lemma shows that this is sufficient to conclude the almost sure convergence Nωn (λ) → N(λ) at continuity points λ, finishing the proof of Theorem 2.6. One has only to observe that every continuous function of compact support on R+ = [0, ∞) can be written as g · f1 , with an element g ∈ A .

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Lemma 4.8 For n ∈ N, let ρ n , ρ be locally finite measures on R+ . Then lim ρ n (g) = ρ(g)

n→∞

for all continuous functions g of compact support implies that lim ρ n ([0, λ]) = ρ([0, λ])

n→∞

for all λ > 0 which are not atoms of ρ. Proof The proof is standard. First note that locally finiteness of ρ implies   lim ρ [λ − ε, λ + ε] = ρ({λ}) = 0. ε→0

Now choose monotone functions gε− , gε+ ∈ Cc (R+ ) satisfying

½[0,λ−ε]  gε−  ½[0,λ]  gε+  ½[0,λ+ε] . Then         ρ ([0, λ]) − ρ n ([0, λ])  ρ gε+ − ρ gε− + ρ gε− − ρ n gε−      ρ ([λ − ε, λ + ε]) + ρ gε− − ρ n gε− . For any δ > 0 one can choose ε > 0 such that ρ([λ − ε, λ + ε]) < δ. Since δ > 0 was arbitrary, we have shown ρ([0, λ])  lim infn→∞ ρ n ([0, λ]). The opposite inequality is shown similarly.  

5 Proof of the Wegner Estimate This section is devoted to the proof of Theorem 2.9. Let (X, , P, ) be a random length model satisfying Assumption 2.8. We first introduce a new measurable map α :  × E −→ [ω− , ω+ ] with ω− = ln min , ω+ = ln max , defined by αω (e) := α(ω, e) = ln ω (e). The random variables α(·, e), e ∈ E, are independently distributed with density functions ge (x) = ex he (ex ), and we have   ge ∞  max he ∞ + 2max h e ∞  max + 2max Ch =: Dh < ∞. (5.1)  Thus, we can re-identify  with the Cartesian product e∈E [ω− , ω+ ], and the maps α(·, e) are simply projections  to the component with index e. The measure P is now given as the product e∈E & Pe of marginal measures & Pe with density 1 functions ge ∈ C (R) satisfying the above estimate (5.1). The advantage of the  new ‘rescaled’ identification  = e∈E [ω− , ω+ ] is the following property of the eigenvalues of the Laplacian on any compact subgraph (, ω ): * +   −2s λi ,D . (5.2) λi ,D ω+s½ = e ω

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Here, the eigenvalues λi are counted with multiplicity and ω + s½ denotes the element {ωe + s}e∈E(X) ∈ . Property (5.2) is an immediate consequence of (2.3b), ω+s½ (e) = eαω (e)+s = es ω (e), and the fact that a rescaling of all lengths by a fixed multiplicative constant does not change the domain of the Kirchhoff Laplacian with Dirichlet boundary conditions on ∂. Property (5.2) is of crucial importance for the proof of the Wegner estimate. Henceforth, we use this new interpretation of  and rename & Pe by Pe , for simplicity. Let  ⊂ X be a compact topological subgraph, λ ∈ R and ε > 0. We write the interval I as [λ − , λ + ] and start with a smooth function ρ : R −→ [−1, 0] satisfying ρ ≡ −1 on (−∞, −ε], 0  ρ  1/ε, ρ ≡ 0 on [ε, ∞). Moreover, we set ρλ (x) = ρ(x − λ). Then we have

½[λ−ε,λ+ε] (x)  ρλ (x + 2ε) − ρλ (x − 2ε) =

2ε

ρλ (x + t) dt.

−2ε

Using the spectral theorem, we obtain Pω,D ([λ

− ε, λ + ε]) = ½



,D [λ−ε,λ+ε] ω



2ε 

  ρλ ,D + t dt, ω

−2ε

and, consequently, tr Pω,D ([λ − ε, λ + ε]) 

2ε

  tr ρλ ,D + t dt. ω

−2ε

 Denote by ((),  P ) the space () = e∈E() [ω− , ω+ ] with probability measure P = e∈E() Pe , and E (·) denote the associated expectation. E(·) means expectation with respect to the full space (, P). Applying expectation yields     E tr P•,D ([λ − ε, λ + ε]) = E tr P•,D ([λ − ε, λ + ε])



2ε



  tr ρλ ,D + t dt dP (ω). ω

(5.3)

() −2ε

Using the chain rule and scaling property (5.2), we obtain  * ++  ∂   ,D     λ,D   d *  ρλ λi ω + t = ρλ λi ω + t s → λi ,D ω+s½  ∂ωe ds s=0 e∈E()       + t λi ,D = −2ρλ λi λ,D  0. ω ω

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Now, we use that [λ − ε, λ + ε] ⊂ Ju = [1/u, u]. Since supp ρλ ⊂ [λ − ε, λ + ε], we derive ⎛ ⎞  ∂     u 0  tr ρλ ,D +t − ⎝ tr ρλ ,D + t ⎠. (5.4) ω ω 2 ∂ωe e∈E()

For e ∈ E(), denote by e the topological subgraph with vertex set Ve := V() and edge set Ee := E() \ {e}. Using the estimate (5.4), we obtain from (5.3)   E tr P•,D ([λ−ε, λ+ε]) u  − 2



2ε ω+ 

e∈E()( ) −2ε ω − e

* + ∂ tr ρλ ,D +t ge (x) dx dt dPe (ω ) (5.5) (ω ,x) ∂ωe

with (ω , x) ∈ (e ) × [ω− , ω+ ] = (). Next, we want to carry out partial integration with respect to x in (5.5). Before doing so, it is useful to observe, for fixed c ∈ [ω− , ω+ ], * + * + * ++ ∂ ∂ * ,D tr ρλ ,D tr ρλ ,D . (ω ,x) + t = (ω ,x) + t − tr ρλ (ω ,c) + t ∂ωe ∂ωe

(5.6)

Using (5.6) and applying partial integration, we obtain  ω   +   * +   ∂ ,D   tr ρ + t g (x) dx λ e (ω ,x)   ∂ωe   ω−

 ge L1

sup

c ∈[ω− ,ω+ ]

 * + * +  ,D ,D  tr ρλ−t (ω ,c ) − tr ρλ−t (ω ,c)  .

(5.7)

For notational convenience, we identify the compact topological graph consisting only of the edge e and its end-points with e, and we denote by e,D c the Dirichlet-Laplacian on the metric graph (e, c ) defined by c (e) = exp(c). Using (4.3) in Lemma 4.3, we conclude that  * + * +    ,D ,D tr ρλ−t (ω ,c) − tr ρλ−t ω e ⊕ e,D   2 |ρ(∞) − ρ(t − λ)| 2dmax  4dmax , c     ,D   for all values c ∈ [ω− , ω+ ]. Consequently, sup tr ρλ−t ,D (ω ,c ) − tr ρλ−t (ω ,c)  in (5.7) can be estimated from above by  * +  e,D   8dmax + tr ρ e,D +t−λ . c + t − λ − tr ρ c

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Note that all eigenfunctions of the Dirichlet operator e,D are explicitly given c sine functions. Therefore, since λ ∈ [1/u + ε, u − ε] and t ∈ [−2ε, 2ε], there is a constant Cu,max > 0, depending only on u, max , such that    tr ρ e,D + t − λ   Cu, , max c for all exp(c) ∈ [min , max ]. This implies ω   +    * +   ∂ ,D    (8dmax + 2Cu, ) g L ([ω ,ω ]) . tr ρ + t g (x) dx λ e max e 1 − + (ω ,x)   ∂ω e   ω−

Plugging this into inequality (5.5), we finally obtain     max E tr P•,D ([λ − ε, λ + ε])  u 4dmax + Cu,max Dh ln 4ε |E()|, min finishing the proof of Theorem 2.9. Acknowledgements The second author is grateful for the kind invitation to the Humboldt University of Berlin which was supported by the SFB 647. NP and OP also acknowledge the financial support of the Technical University Chemnitz.

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Math Phys Anal Geom (2009) 12:255–286 DOI 10.1007/s11040-009-9061-3

Localization for a Matrix-valued Anderson Model Hakim Boumaza

Received: 10 February 2009 / Accepted: 18 June 2009 / Published online: 27 June 2009 © Springer Science + Business Media B.V. 2009

Abstract We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on L2 (R) ⊗ C N , for arbitrary N ≥ 1. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval I ⊂ R, they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters. Keywords Anderson localization · Lyapunov exponents · Multiscale analysis Mathematics Subject Classifications (2000) 47B80 · 37H15

H. Boumaza (B) Department of Mathematics, Keio University, Hiyoshi 3-14-1, Kohoku-ku 223-8522, Yokohama, Japan e-mail: [email protected]

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1 Introduction: Models and Results In this paper, we will discuss localization properties of continuous matrixvalued Anderson models of the form:  d2 H(ω) = − 2 ⊗ IN + Vω(n) (x − n), (1) dx n∈Z acting on L2 (R) ⊗ C N , where N ≥ 1 is an integer, IN is the identity matrix of order N and  > 0 is a real number. Let (, A, P) be a complete probability space and let ω ∈ . For every n ∈ Z, the functions x → Vω(n) (x) are symmetric matrix-valued functions, supported on [0, ] and bounded uniformly on x, n and ω. The sequence (Vω(n) )n∈Z is a sequence of independent and identically distributed (i.i.d) random variables on . We also assume that the potential  x → n∈Z Vω(n) (x − n) is such that the operator H(ω) is Z-ergodic. d2 As a bounded perturbation of − dx 2 ⊗ IN , the operator H(ω) is self-adjoint on the Sobolev space H 2 (R) ⊗ C N and thus, for every ω ∈ , the spectrum of H(ω) is included in R. Due to the hypothesis of Z-ergodicity, there exists  ⊂ R such that, for Palmost every ω ∈ ,  = σ (H(ω)). There also exist pp , ac and sc , subsets of R, such that, for P-almost every ω ∈ , pp = σpp (H(ω)), ac = σac (H(ω)) and sc = σsc (H(ω)). We will show that under suitable assumptions on the Fürstenberg group of H(ω) (see Definition 4), this operator will exhibit localization properties on a certain interval of R. These assumptions are not satisfied for every operators of the form (1), but we will verify them for the following operator: ⎛ ⎞ 0 c1 ω1(n) 1[0,] (x − n) 2  d ⎜ ⎟ .. H (ω) = − 2 ⊗ IN +V0 + ⎝ ⎠, . dx (n) n∈Z 0 c N ω N 1[0,] (x − n) (2) acting on L2 (R) ⊗ C N . The real number  > 0 represents the length of the range of the random interactions. The constants c1 , . . . , c N are non-zero real numbers and V0 is the multiplication operator by the tridiagonal matrix V0 having a null diagonal and coefficients on the upper and lower diagonals all equal to 1. For every i ∈ {1, . . . , N}, the (ωi(n) )n∈Z are sequences of i.i.d. random vari ,

, A ables on a complete probability space ( P), of common law ν such that {0, 1} ⊂ supp ν and supp ν is bounded. The random parameter ω is an element of the product space 

⊗N , ⊗n∈Z

⊗N , ⊗n∈Z A (, A, P) = ⊗n∈Z  P⊗N ⊗N . The and we also set, for every n ∈ Z, ω(n) = (ω1(n) , . . . , ω(n) N ), of law ν expectancy against P will be denoted by E(.).

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The model (2) is a particular case of (1). Indeed, H (ω) is Z-ergodic and the potential part of H (ω) is uniformly bounded on x, n and ω because of the boundedness of supp ν. Following [16], we give the definitions of localization properties for H(ω), from the spectral and the dynamical point of views. For x ∈ R, we denote by 1x the characteristic

function of the interval of length 2 centered at x. We also write < x >= 1 + |x|2 and we denote by Eω (.) the spectral projection of H(ω). The Hilbert-Schmidt norm is written as  2 while the L2 -norm is written as  . Definition 1 Let I ⊂ R be an open interval. We say that: (i)

H(ω) exhibits exponential localization (EL) in I, if it has pure point spectrum in I (i.e.,  ∩ I = pp ∩ I and ac ∩ I = sc ∩ I = ∅) and, for P-almost every ω, the eigenfunctions of H(ω) with eigenvalues in I decay exponentially in the L2 -sense (i.e., there exist C and m > 0 such that 1x ψ ≤ Ce−m|x| for ψ eigenfunction of H(ω)); (ii) H(ω) exhibits strong dynamical localization (SDL) in I, if  ∩ I = ∅ and, for each compact interval I˜ ⊂ I and ψ ∈ L2 (R) ⊗ C N with compact support, we have,  2    n   ∀n ≥ 0, E sup < x > 2 Eω I˜ e−itH(ω) ψ  < ∞ ; t∈R

(iii)

H(ω) exhibits strong sub-exponential HS-kernel decay (SSEHSKD) in I if  ∩ I = ∅ and, for each compact interval I˜ ⊂ I and 0 < ζ < 1, there is a finite constant C I,ζ ˜ such that,  ∀x, y ∈ Z, E

 2    sup 1x Eω I˜ f (H(ω))1 y 

 f ∞ ≤1

2

 ζ

−|x−y| ≤ C I,ζ , ˜ e

f being a bounded Borel function on R and  f ∞ = supt∈R | f (t)|. We also set EL , SDL and SSEHSKD as the sets of E ∈  for which there exists an open interval I, E ∈ I, such that H(ω) exhibits on I, (EL), (SDL) and (SSEHSKD) respectively. We have SSEHSKD ⊂ SDL and we will actually prove (SSEHSKD) for H (ω) on some interval, which will imply (SDL) on the same interval. We quote (SDL) property as it has a more natural interpretation than (SSEHSKD) in terms of control of the moments of the wave packets of H(ω). We are now ready to give the statement of our main results. For E ∈ R, let G(E) be the Fürstenberg group associated to H(ω) (see Definition 4). For the definitions of p-contractivity and L p -strong irreducibility, see Definition 3.

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Theorem 1 Let I ⊂ R be a compact interval such that  ∩ I = ∅ and let I˜ be an ˜ such that, for every E ∈ I, ˜ G(E) is p-contracting and Lp open interval, I ⊂ I, strongly irreducible, for every p ∈ {1, . . . , N}. Then, H(ω) exhibits (EL), (SDL) and (SSEHSKD) in I. Before applying this theorem to the operator H (ω), we need to introduce some notations. Let SpN (R) denote the group of 2N × 2N real symplectic matrices and let O be the neighborhood of I2N in SpN (R) given by Theorem 6 applied to G = SpN (R) . We set: dlog O = max{R > 0 | B(0, R) ⊂ log O}, where B(0, R) is the open ball, centered on 0 and of radius R > 0, for the topology induced on the Lie algebra spN (R) of SpN (R) by the matrix norm induced by the euclidean norm on R2N . N For ω(0) = (ω1(0) , . . . , ω(0) N ) ∈ {0, 1} , let  Mω(0) = V0 + diag c1 ω1(0) , . . . , c N ω(0) N . (0)

(0)

As Mω(0) is a real symmetric matrix, it has λω1 , . . . , λωN as real eigenvalues. We set, λmin = and δ0 =

λmax −λmin . 2

min

ω(0) ∈{0,1} N

(0)

min λiω ,

1≤i≤N

λmax =

max

ω(0) ∈{0,1} N

max λiω

1≤i≤N

(0)

(3)

We also set 

dlog O C := C (N) = min 1, δ0

 (4)

and, for every  < C ,

  dlog O dlog O , λmin + . I(, N) = λmax −  

(5)

Applying Theorem 1 to the operator H (ω), we obtain the following results. Theorem 2 (i) Assume that  < C and let I ⊂ I(, N) be an open interval such that  ∩ I = ∅. Then H (ω) exhibits (EL), (SDL) and (SSEHSKD) in I. (ii) Assume that  = 1 and N = 2 in (2). There exists a discrete set S ⊂ R such that, for every compact interval I ⊂ (2, +∞) \ S with  ∩ I = ∅, H1 (ω) exhibits (EL), (SDL) and (SSEHSKD) in I. We remark that in point (i) of Theorem 2, as the length of I(, N) tends to +∞ when  tends to 0+ , taking  small enough ensure that  ∩ I(, N) = ∅ and, moreover, we can always find a non-trivial open interval I ⊂ I(, N) such that  ∩ I = ∅.

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To prove localization results as Theorems 1 and 2 for one-dimensional operators such as (1) and (2), we can follow this plan: 1. We prove that the Lyapunov exponents of H(ω) are all distinct and positive. 2. We prove the Hölder regularity of these exponents. 3. We deduce the same Hölder regularity for the integrated density of states of H(ω). 4. With this regularity of the integrated density of states, we prove a Wegner estimate. 5. We apply a multiscale analysis scheme. According to this plan, our first result for H (ω) is the following. For the definitions of μ E and L p , see Section 2.1. Theorem 3 Assume that  < C . Then, (i) the N positive Lyapunov exponents of H (ω), γ1 (E), . . . , γ N (E), verify ∀E ∈ I(, N), γ1 (E) > · · · > γ N (E) > 0.

(6)

Therefore, H (ω) has no absolutely continuous spectrum in I(, N), i.e., ac ∩ I(, N) = ∅. (ii) For every p ∈ {1, . . . , N}, there exists a unique μ E -invariant measure ν p,E on P(L p ) = {x¯ ∈ P(∧ p R2N ) | x ∈ L p } such that, for every E ∈ I(, N),  p   (∧ p M)x γi (E) = log (7) dμ E (M)dν p,E x¯ . x SpN (R) ×P(L p ) i=1 (iii) For every i ∈ {1, . . . , N}, E → γi (E) is Hölder continuous on I(, N), i.e., there exist C > 0 and α > 0 such that,   α   (8) ∀E, E ∈ I(, N), γi (E) − γi E  ≤ C E − E  . Points (i) and (ii) will directly follow from the theory of sequences of i.i.d. random matrices in SpN (R) after proving that, for every E ∈ I(, N), the Fürstenberg group of H (ω) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. It is exactly the assumption of Theorem 1. Then, applying results of [3], we obtain a regularity result for the integrated density of states E → N(E). Theorem 4 Let  < C . Let I be a compact interval included in the interior of I(, N). The integrated density of states of H (ω) is Hölder continuous on I, i.e., there exist C > 0 and α > 0 such that,   α   (9) ∀E, E ∈ I,  N(E) − N E  ≤ C E − E  . The local Hölder regularity of the integrated density of states is a key ingredient to prove a Wegner estimate for H(ω). Let L ∈ N∗ and denote

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by H (L) (ω) the restriction of H(ω) to L2 ([−L, L]) ⊗ C N with Dirichlet boundary conditions. We define H(L) (ω) the same way, for every  > 0. ˜ Theorem 5 Let I ⊂ R be a compact interval and I˜ be an open interval, I ⊂ I, ˜ such that, for every E ∈ I, G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Then, for every β ∈ (0, 1) and every κ > 0, there exist L0 ∈ N and ξ > 0 such that,    β β P d E, σ (H (L) (ω)) ≤ e−κ(L) ≤ e−ξ(L) , (10) for every E ∈ I and every L ≥ L0 . Applying this general result to H (ω) and using Theorem 3, lead us to the following. Corollary 1 (i) Let  < C . Then (10) holds for H(L) (ω) for every E ∈ I, where I is a nontrivial interval included in the interior of I(, N). (ii) Let  = 1 and N = 2 in (2). There exists a discrete set S ⊂ R such that, for every compact interval I ⊂ (2, +∞) \ S , (10) holds for H1(L) (ω), for every E ∈ I. Then, to obtain Theorems 1 and 2, it will remain to show that we can apply a multiscale analysis scheme as presented in [20] or [16]. After [10], we already know that, in the scalar-valued case (corresponding here to N = 1), there exists a discrete set S ⊂ R such that, on every compact interval I ⊂ R \ S ,  ∩ I = ∅, we have exponential localization and strong dynamical localization. Thus, there is localization at small and large energies in this case. On higher dimension d ≥ 1, the multi-dimensional Anderson model is defined by the operator  H(ω) = −d + Vω(n) (x − n), (11) n∈Zd

acting on L2 (Rd ) ⊗ C, with the same assumptions as for model (1), in the case N = 1. In particular, we are interested in handling singular distributions of the random parameter ω. For d ≥ 2, it is known since [6], that there is spectral localization near the bottom of the almost-sure spectrum of H(ω). There is not yet any proof of dynamical localization for this model. For large energies, the question of the localization is still open, under any assumption on the distribution of the random variables. It is commonly conjectured that, for d ≥ 3, there exist delocalized states at large energies. On the contrary, for d = 2, it is conjectured that there is localization, even for large energies, exactly like in the case d = 1. To tackle the question of localization for d = 2, including singular distribution of ω, we can start by studying a simpler model, a continuous strip in R2 . We consider the restriction of (11) to the continuous strip R × [0, 1],

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acting on L2 (R × [0, 1]) ⊗ C, with Dirichlet boundary conditions on R × {0} and R × {1}. This continuous strip model is not only interesting as a first step to study localization on R2 , but it is also of physical interest. Indeed, such a model can be considered to modelize nanoconductors and it allows to study their transport properties. The question of localization at all energies for the continuous strip R × [0, 1] is a difficult problem, mostly due to its PDE’s nature. A possible approach for this question is to operate a discretization in the bounded direction. Such a discretization can be performed by first applying discrete Fourier transform in the bounded direction of the strip. It leads to a quasi-one dimensional model with a matrix-valued potential of infinite size (corresponding to N = ∞ in (1)). Then, to obtain the desired discretization, we consider an arbitrary large cutoff in the space of Fourier frequencies. Thus, we are bringed back to model (1) with finite and arbitrary large N. It turns the nature of the initial PDE’s problem to an ODE’s one, which allows to use formalism such as transfer matrices and Lyapunov exponents. Then, we want to obtain localization for arbitrary large N and we hope to be able to recover localization properties, first for the infinite order quasi-one dimensional model in the Fourier space, by letting N “tends to infinity”, and then for the continuous strip in dimension 2. For this purpose, it is important to have results for arbitrary N ≥ 1. In a previous article of the author, [2], we proved separability of the Lyapunov exponents of H1 (ω) for large energies, but only for N = 2. It was done by proving p-contractivity and L p -strong irreducibility of the Fürstenberg group for energies E > 2 (for N = 2, λmax = 2) and away from a discrete set of R. Point (ii) of Theorem 2 is based upon this result. Due to some technical difficulties, it was not possible to generalize the computations done for N = 2 to an arbitrary N ≥ 1. This is were the parameter  > 0 play an important role, as explained in the end of Section 2.2. The main difference between what was proved for  = 1 and N = 2 in [2], and the case N ≥ 1 for  small considered here, is the existence of a discrete set of critical energies which does not appear in the second case. We also want to mention that different methods have been used in [15] to prove localization for random operators on strips. These methods, using spectral averaging techniques, do not apply for singular distribution of the random parameters. We choose instead to follow methods of [17] for the discrete strip and adapt them to our models. The same strategy was already followed in [10] for the scalar-valued case. We finish this introduction by giving the outline of the rest of the article. In Section 2, we prove Theorem 3. We first recall definitions of the Lyapunov exponents in Section 2.1 before introducing the Fürstenberg group of H (ω) and study it in Section 2.2. In Section 3, we review and adapt estimates on the random walk defined by the transfer matrices and, in particular, large deviation type estimates. Then we shortly discuss the Hölder regularity of the integrated density of states in Section 4. We can deduce from this regularity result a Wegner estimate, as it is done in Section 5. Finally, in Section 6, we give the proofs of Theorem 1 and of Theorem 2. We start by presenting the

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requirements of a multiscale analysis scheme in Section 6.1 and then we prove, in Section 6.2, an initial length scale estimate required to apply this scheme.

2 Positivity and Regularity of the Lyapunov Exponents In this Section, we will give the proof of Theorem 3. Before that, we will set up notations and definitions about Lyapunov exponents. 2.1 Lyapunov Exponents We start with a review of the definition of the Lyapunov exponents. Let N be a positive integer and let SpN (R) denote the group of 2N × 2N real symplectic matrices. It is the subgroup of GL2N (R) of matrices M satisfying t

MJM = J,

where J is the matrix of order 2N defined by J =



 0 −IN . IN 0

Definition 2 Let (Tnω )n∈N be a sequence of i.i.d. random matrices in SpN (R) with E(log+ T0ω ) < ∞. The Lyapunov exponents γ1 , . . . , γ2N associated with (Tnω )n∈N are defined inductively by p 

γi = lim

i=1

n→∞

  ω  1  E log  ∧ p Tn−1 . . . T0ω  , n

(12)

for every p ∈ {1, . . . , 2N}. Here, ∧ pM denotes the pth exterior power of the matrix M, acting on the pth exterior power of R2N . One has γ1 ≥ . . . ≥ γ2N . Moreover, we have the symmetry property γ2N−i+1 = −γi , for every i ∈ {1, . . . , N}, due to the symplecticity of the random matrices Tnω . We will only have to study the N first Lyapunov exponents, those being positive. We also define, for every p ∈ {1, . . . , N}, the p-Lagrangian submanifold Lp of R2N , as the subspace of ∧ p R2N spanned by {Me1 ∧ . . . ∧ Me p | M ∈ SpN (R) }, where (e1 , . . . , e2N ) is the canonical basis of R2N . We note that L1 = ∧1 R2N = R2N . We can now give the definitions of p-contractivity and L p - strong irreducibility. Definition 3 Let G be a subset of SpN (R) and p ∈ {1, . . . , N}. (i) G is p-contracting if there exists a sequence (Tn )n∈N in G such that the sequence ( ∧ p Tn −1 ∧ p Tn )n∈N converges to a rank-one matrix. (ii) G is L p -strongly irreducible if there does not exist a finite union W of proper subspaces of L p such that, (∧ p T)(W) = W for any T ∈ G. We can now give the proof of Theorem 3.

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2.2 Proof of Theorem 3 In this Section, all the definitions are given for the operator H (ω), but we could also define the same objects for the more general operator (1). We start by introducing the sequence of transfer matrices associated to the operator H (ω). Let E ∈ R. The transfer matrix Tω(n) (E) of H (ω), from n to (n + 1), is defined by the relation     u((n + 1)) u(n) ∀n ∈ Z, = Tω(n) (E) , (13) u ((n + 1)) u (n) where u : R → C N is a solution of the second order differential system H (ω)u = Eu.

(14)

We can give the explicit form of the matrices Tω(n) (E). For E ∈ R, n ∈ Z and ˜ ⊗N , we set ω(n) ∈   Mω(n) (E) = V0 + diag c1 ω1(n) , . . . , c N ω(n) (15) N − EIN . Then, if we set

 Xω(n) (E) =

IN Mω(n) (E) 0 0

 ∈ M2N (R),

(16)

by solving the constant coefficient system (14) on [n, (n + 1)], we have: ∀ > 0, ∀n ∈ Z, ∀E ∈ R, Tω(n) (E) = exp (Xω(n) (E)) .

(17)

The fact that Tω(n) (E) is the exponential of a matrix will be very important to be able to apply Theorem 6. We can now introduce the central object involved in the proof of Theorem 3. It is the algebraic object containing all the products of transfer matrices. Definition 4 For every real number E ∈ R, the Fürstenberg group of H (ω) is defined by G(E) = < supp μ E >, where μ E is the common distribution of the Tω(n) (E). As the Tω(n) (E) are i.i.d., μ E = (Tω(0) (E))∗ ν ⊗N and we have the internal description of G(E): ∀E ∈ R, G(E) = < Tω(0) (E) | ω(0) ∈ supp ν ⊗N >.

(18)

As {0, 1} ⊂ supp ν, we also have G(E) ⊃ < Tω(0) (E) | ω(0) ∈ {0, 1} N >. Due to a criterion of Gol’dsheid and Margulis (see [5, 14]), to prove that, for a given E ∈ R, G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}, it suffices to prove that G(E) is Zariski dense in SpN (R) . Actually, we will prove a stronger statement which is that, for every E ∈ I(, N), G(E) is equal to SpN (R) . Therefore, for every E ∈ I(, N), G(E) will be p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}.

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Proposition 1 Let  < C and I(, N) be the interval defined at (5). Then, for every E ∈ I(, N), G(E) = SpN (R) . The proof of this proposition is based upon the following theorem due to Breuillard and Gelander. Theorem 6 ([7], Theorem 2.1) Let G be a real, connected, semisimple Lie group, whose Lie algebra is g. Then, there is a neighborhood O of 1 in G, on which log = exp−1 is a well defined diffeomorphism, such that g1 , . . . , gm ∈ O generate a dense subgroup whenever log g1 , . . . , log gm generate g. This theorem gives us the outline of the proof of Proposition 1: 1. We prove that, for every  ∈ (0, C ) and every E ∈ I(, N), Tω(0) (E) ∈ O, for every ω(0) ∈ {0, 1} N . 2. For  < C , we compute log Tω(0) (E). 3. We prove that Lie{log Tω(0) (E) | ω(0) ∈ {0, 1} N } = spN (R) , the Lie algebra of SpN (R) . Before proving Proposition 1, we prove the following algebraic lemma which will be used to prove point 3. Lemma 1 Let N ≥ 1 and E ∈ R. The Lie algebra generated by {Xω(0) (E) | ω(0) ∈ {0, 1} N } is equal to spN (R) . Proof First, we recall that:    a b1 spN (R) = M ( R ), b and b symmetric . , a ∈ N 1 2 b2 −t a For i, j ∈ {1, . . . , N}, let Eij be the matrix in MN (R) with a 1 coefficient at the intersection of the ith row and the jth column, and 0 elsewhere. We also set     1 0 Eij + E ji Eij 0 t , Yij = Xij, Z ij = . ∀i, j ∈ {1, . . . , N}, Xij = 0 0 −E ji 2 0 We also denote by δij the Kronecker’s symbol:  1 if i = j δij = 0 if i = j. We remark that the set {Xij, Yij, Z ij}i, j=1..N is a basis of spN (R) . By direct computation, we get the relations, for every i, j, k, r ∈ {1, . . . , N}, (i) [Z ij, Xkr ] = δ jk Xir + δ jr Xik (ii) [Ykr , Z ij] = δik Yrj + δir Ykj (iii) [Xij, Ykr ] = 14 (δ jk Z ir + δ jr Z ik + δki Z jr + δir Z jk )

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where [ , ] is the usual bracket on Lie algebra of linear Lie groups. From these relations, we deduce that spN (R) is generated by   Xij, Yij | i, j ∈ {1, . . . , N}, |i − j| ≤ 1 . Indeed, let g be the Lie algebra generated by this set. Let i ∈ {1, . . . , N}. Then, Z ii = 2[Xii , Yii ] ∈ g and Z i,i+1 = 2[Xii , Yi,i+1 ] ∈ g. Thus, for every i, j ∈ {1, . . . , N}, |i − j| ≤ 1, Z ij ∈ g. Then we have, Xi,i+2 = [Z i,i+1 , Yi+1,i+2 ], Yi,i+2 = [Yi,i+1 , Z i+1,i+2 ] ∈ g and Z i,i+2 = 2[Xi,i+1 , Yi+1,i+2 ] ∈ g. Thus, for every i, j ∈ {1, . . . , N}, |i − j| = 2, Xij, Yij, Z ij ∈ g. By induction, we do the same for indices i, j such that |i − j| = 3 and more generally for all indices i, j ∈ {1, . . . , N}. Thus, we proved that {Xij, Yij, Z ij}i, j=1..N is included in g and then spN (R) ⊂ g. Finally, g = spN (R) . According to this, to prove Lemma 1, we only have to prove that, for every E ∈ R, the Lie algebra generated by {Xω(0) (E) | ω(0) ∈ {0, 1} N } contains all the matrices Xij and Yij for i, j ∈ {1, . . . , N}, |i − j| ≤ 1. Let    (19) a(E) = Lie Xω(0) (E)  ω(0) ∈ {0, 1} N . To prove that a(E) contains the matrices Xij and Yij for i, j ∈ {1, . . . , N}, |i − j| ≤ 1, we will proceed in several steps. We fix E ∈ R. Step 1 We prove that the matrices Z ii for i ∈ {1, . . . , N} are in a(E). Let ω(0) and ω˜ (0) in {0, 1} N . We have: [Xω(0) (E), Xω˜ (0) (E)] = Xω(0) (E)Xω˜ (0) (E) − Xω˜ (0) (E)Xω(0) (E)    (0) = diag c1 ω˜ 1(0) − ω1(0) , . . . , c N ω˜ (0) N − ωN ,    (0) c1 ω1(0) − ω˜ 1(0) , . . . , c N ω(0) − ω ˜ . N N In particular, for ω(0) = (0, . . . , 0) and ω˜ (0) = (0, . . . , 1, . . . , 0), with a 1 at the ith place and 0 elsewhere, we get Z ii = [Xω(0) (E), Xω˜ (0) (E)] ∈ a(E). Step 2 With the same choice of ω(0) and ω˜ (0) , we get Xω˜ (0) (E) − Xω(0) (E) = Yii . Thus, for every i ∈ {1, . . . , N}, Yii ∈ a(E). Step 3 We fix ω(0) ∈ {0, 1} N and i ∈ {1, . . . , N}. We have:   0 −2Eii [Xω(0) (E), Z ii ] = Mω(0) (0)Eii + Eii Mω(0) (0) − 2EEii 0 = −2Xii + 2Yi,i−1 + 2Yi,i+1 + 2(ωi(0) − E)Yii with the convention that Yij is zero if the index j is not in {1, . . . , N}. Thus, dividing by 2, one gets,  ∀i ∈ {1, . . . , N}, −Xii + Yi,i−1 + Yi,i+1 + ωi(0) − E Yii ∈ a(E). (20)

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Step 4 We prove that the matrix J is in a(E). We fix ω(0) = (0, . . . , 0). By summing (20) for i ∈ {1, . . . , N}, we stay in a(E) and we have:   N N    0 0 (−Xii ) + −Xii +Yi,i−1 +Yi,i+1 − EYii = ∈ a(E). Mω(0) (E) 0 i=1

i=1

We can subtract Xω(0) (E) ∈ a(E) from this, to get:     N  0 −2IN 0 −IN = ∈ a(E). (−Xii ) + 0 0 0 0 i=1   0 −IN Thus, ∈ a(E). But, by Step 2, all the Yii ’s are in a(E), so we 0 0 also have:   N  0 0 Yii = ∈ a(E). IN 0 i=1   0 −IN By adding these two matrices, J = ∈ a(E). IN 0 Step 5 For every i ∈ {1, . . . , N}, [J, Z ii ] = 2Yii + 2Xii ∈ a(E). But Yii ∈ a(E), so 2Xii = [J, Z ii ] − 2Yii ∈ a(E) and, for every i ∈ {1, . . . , N}, Xii ∈ a(E). Step 6 We recall that Xij = X ji and Yij = Y ji . Let i ∈ {1, . . . , N}. Subtracting (ω(0) − E)Yii ∈ a(E) and adding Xii ∈ a(E) in (20) we get Yi,i−1 + Yi,i+1 ∈ a(E). For i = 1, it means that Y1,2 ∈ a(E). Then, 1 Z = [X1,1 , Y1,2 ] ∈ a(E) and Z 1,2 ∈ a(E). But we also have 2X1,2 = 2 1,2 [Z 1,2 , X2,2 ] ∈ a(E) and X1,2 ∈ a(E). Now, for i = 2, we have Y2,1 + Y2,3 ∈ a(E). But we just proved that Y2,1 ∈ a(E), thus Y2,3 ∈ a(E). Inductively, we prove that: ∀i ∈ {1, . . . , N}, Yi,i+1 ∈ a(E). Also, for every i ∈ {1, . . . , N}, 1 [Xii , Yi,i+1 ] = Z i,i+1 ∈ a(E) and [Z i,i+1 , Xi+1,i+1 ] = 2Xi,i+1 ∈ a(E). 2 It proves that all the matrices Xij and Yij for i, j ∈ {1, . . . , N} and |i − j| ≤ 1 are in a(E). Thus, a(E) = spN (R) .   We can now prove Proposition 1. Proof (of Proposition 1) In this proof we directly construct C and I(, N). Let (0) (0) λω1 ,. . ., λωN be the real eigenvalues of the real symmetric matrix Mω(0) (0), as in (0) the introduction. Then, the eigenvalues of Xω(0) (E)t Xω(0) (E) are 1, (λω1 − E)2 , (0) . . ., (λωN − E)2 . Thus:   ω(0) Xω(0) (E) = max 1, max |λi − E| , 1≤i≤N

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where   is the matrix norm associated to the euclidian norm on R2N . Let O be the neighborhood of the identity given by Theorem 6 for G = SpN (R) . Then O depends only on N. To apply Theorem 6 to G(E) ⊂ SpN (R) , we need to find an interval of values of E such that, for  small enough, ∀ω(0) ∈ {0, 1} N , 0 < Xω(0) (E) < dlog O , or, equivalently,

 0 <  max 1,

max

 (0) max |λiω − E| < dlog O .

ω(0) ∈{0,1} N 1≤i≤N

(21)

(22)

We assume that  ≤ dlog O and we set r = 1 dlog O ≥ 1. We want to characterize the set:       (0) (23) I(, N) = E ∈ R  max 1, max max |λiω − E| ≤ r . ω(0) ∈{0,1} N 1≤i≤N

As r ≥ 1,



I(, N) =

 

 (0) (0) λiω − r , λiω + r .

(24)

ω(0) ∈{0,1} N 1≤i≤N

Let λmin , λmax and δ0 be as in (3). If δ0 < r , I(, N) = ∅ and more precisely, max I(, N) = [λmax − r , λmin + r ]. This interval is centered in λmin +λ and is of 2 length 2r − 2δ0 > 0. Moreover, 2r − 2δ0 → +∞ when  tends to 0+ . As λmin , λmax and dlog O depend only on N, I(, N) depends only on  and N and the condition δ0 < r is equivalent to <

dlog O = C (N). δ0

So, we have just proved that, ∀ < C , ∀E ∈ I(, N), 0 < Xω(0) (E) ≤ dlog O .

(25)

Thus, for every E ∈ I(, N), log Tω(0) (E) = Xω(0) (E), as exp is a diffeomorphism from log O into O. Then, we can apply Lemma 1 to obtain:   ∀ > 0, ∀E ∈ R, Lie Xω(0) (E) | ω(0) ∈ {0, 1} N = spN (R) . (26) Applying Theorem 6, we get: ∀ < C , ∀E ∈ I(, N), < Tω(0) (E) | ω(0) ∈ {0, 1} N > = SpN (R) .

(27)

As < Tω(0) (E) | ω(0) ∈ {0, 1} N > ⊂ G(E) and G(E) ⊂ SpN (R) , we finally have ∀ < C , ∀E ∈ I(, N), G(E) = SpN (R) , which proves Proposition 1. We can finally prove Theorem 3.

(28)  

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Proof (of Theorem 3) As, for every  < C and every E ∈ I(, N), G(E) = SpN (R) , we get that, for every  < C and every E ∈ I(, N), G(E) is pcontracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Then, by [1, Proposition IV.3.4], we get the separability and positivity of the Lyapunov exponents and their integral representation (7), together with the existence of the μ E -invariant measure ν p,E , for every p ∈ {1, . . . , N}. For the assertion on absence of absolutely continuous spectrum, we refer to Kotani’s theory [18]. For the regularity result (8), we can directly apply [3, Theorem 2] on the interval I(, N).   In [2] we also used Theorem 6 to obtain the separability of the Lyapunov exponents for the model studied there. This model corresponds to the case N = 2 and  = 1 of H (ω). The main difference between [2] and the proof we have just given is that in [2] we could not let  get small and then just control E, to ensure that Xω(0) (E) ∈ log O and thus Tω(0) (E) ∈ O, uniformly on ω(0) ∈ {0, 1} N . We had first used simultaneous diophantine approximation to find a suitable power of Tω(0) (E), say (Tω(0) (E))mω(0) (E) , which is in O. Then arised difficulties with the computations of the logarithm. First, log(exp(mω(0) (E)Xω(0) (E))) = mω(0) (E)Xω(0) (E) as mω(0) (E)Xω(0) (E) ∈ / log O in general. It leads to a problem of determination of the logarithm and the existence of a discrete set of critical energies S such that, for E ∈ S , log(Tω(0) (E))mω(0) (E) is not defined. Then, for E∈ / S , the expression of these logarithms being not simple, we could not use an algebraic result like Lemma 1 to prove that the Lie algebra generated by the logarithms is spN (R) . That is why we had to restrict ourselves to N = 2 in [2].

3 Estimates on the Products of Transfer Matrices In this Section, we review and adapt results precising the convergence of the sequence (∧ pU (n) (E))n∈Z where, for every n ∈ Z, U (n) (E) = Tω(n−1) (E) · · · Tω(0) (E). In particular, we will prove large deviation type estimates. Let I ⊂ R be an open interval such that, for every E ∈ I, G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Lemma 2   (∧ pU (n) (E))x 1 E log −−−−→ γ1 (E) + . . . + γ p (E), n→+∞ n x uniformly in E ∈ I and x¯ ∈ P(L p ). For the proof of this lemma, we refer to [9, Proposition V.4.9 ] or [4, Proposition 6.3.4].

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Lemma 3 Let p ∈ {1, . . . , N}. There exist ξ1 > 0, δ > 0 and n1 ∈ N such that, for every E ∈ I, n ≥ n1 and x ∈ ∧ p (R2N ), x = 1, we have:  −δ  E  ∧ p U (n) (E)x (29) ≤ e−ξ1 n . Proof We fix p ∈ {1, . . . , N}. We set NU =  ∧ p U (n) (E)x. We start by writing NU−δ = e−δ log NU , and using the inequality e y ≤ 1 + y + y2 e|y| , for any y ∈ R. Then, for every E ∈ I and every δ > 0,   (30) E NU−δ ≤ 1 − δ E (log NU ) + δ 2 E (log NU )2 eδ log NU . But, as eδ log NU = NUδ , x = 1 and the Tω(n) (E) are i.i.d., by Cauchy–Schwarz inequality,   1  1 E (log NU )2 eδ log NU ≤ E (log NU )4 2 E NU2δ 2 ⎡ & 12 4 ⎤ 12 $n−1 n−1  % ≤ E⎣ p log Tω(i) (E) ⎦ E Tω(i) (E)2 pδ i=0

i=1

 1  n ≤ n2 p2 E (log Tω(0) (E))4 2 E Tω(0) (E)2 pδ 2 . Thus, there exist constants C1 = C1 (I) and C2 = C2 (I) such that,  −δ      n ≤ 1 − δ E log  ∧ p U (n) (E)x + δ 2 n2 C1 C2 . E  ∧ p U (n) (E)x

(31)

If C > 0 is such that, for every E ∈ I, Tω(0) (E) ≤ C, we can choose C1 = ( p log C)2 and C2 = C pδ . Moreover, by Lemma 2, there exist n0 ≥ 1, uniform in E ∈ I, and x normalized, such that,   1 E  ∧ p U (n0 ) (E)x−δ ≤ 1 − n0 δ γ1 (E) + · · · + γ p (E) + δ 2 n20 C1 C2n0 2 ≤ 1 − ε, for ε > 0, if we choose δ small enough. Then, for n ≥ 1, we set [ nn0 ] the largest integer less than or equal to nn0 , and we write the euclidian division of n by n0 , n = [ nn0 ]n0 + r, 0 ≤ r < n0 . Then, there exist n1 ≥ 1, a constant C˜ and ξ1 > 0 such that for every p ∈ {1, . . . , N},  −δ  n ˜ − ε)[ n0 ] ≤ e−ξ1 n . ∀n ≥ n1 , ∀E ∈ I, E  ∧ p U (n0 ) (E)x ≤ C(1 For this, we refer to [8, Lemma 5.1].

 

This lemma will be used in the proof of a Wegner estimate for H (ω). Later, we will need results on large deviation for the random walk (U (n) (E))n∈Z . We

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will now briefly quote them, following or refering to [1] and [10] for the proofs. The first result is an estimate on the μ E -invariant measure ν p,E introduced at Theorem 3. Proposition 2 Let p ∈ {1, . . . , N}. Let δ(x¯ , y¯ ) be the projective distance between x¯ and y¯ on P(L p ). We assume that E ∈ R is such that G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Then, there exist ρ > 0 and C > 0 such that, for every x¯ ∈ P(L p ) and every ε > 0,   ν p,E y¯ ∈ P(L p ) | δ(x¯ , y¯ ) ≤ ε ≤ Cερ . (32) Proof It comes from a simple adaptation to the symplectic case of Theorem VI.2.1 and Proposition VI.4.1 in [1]. Thus, we deduce this result as in Corollary VI.4.2 in [1].   Lemma 4 Let p ∈ {1, . . . , N}. We assume that E ∈ R is such that G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Then, there exists κ0 > 0 such that, for every ε > 0, x ∈ L p , x = 0, lim sup n→+∞

  1 log P log (∧ pU (n) (E))x − n(γ1 + · · · + γ p )(E) > nε < −κ0 . n (33)

Proof We refer to [1, Theorem V.6.2], replacing Sn there by ∧ pU (n) (E) here   and γ by (γ1 + · · · + γ p )(E). Lemma 5 Let p ∈ {1, . . . , N}. We assume that E ∈ R is such that G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Let y ∈ L p of norm y = 1. For every ε > 0, there exist κ1 > 0 and n0 ∈ N such that,   |((∧ pU (n) (E))x, y)| −εn <e ∀n ≥ n0 , sup P < e−κ1 n . (34) p U (n) (E))x (∧ x∈L p ,x =0 Proof We can directly adapt [10, Lemma 6.3] by replacing fn there by fn : [0, ] → R, ⎧ if 0 ≤ t ≤ e−εn ⎨1 εn fn (t) = 2 − te if e−εn ≤ t ≤ 2e−εn ⎩ 0 if 2e−εn ≤ t ≤  for ε sufficiently small, and P(R2 ) by P(L p ) in the definition of n . We also use an estimate on the convergence to ν p,E of the sequence of powers of the image of μ E by the Fourier-Laplace operator (see [9, Corollary V.4.12], or [4, Proposition 6.3.13(iii)]). In particular, at (6.6) of [10], we assume that log ρ + ε < 0 and we set δ1 = 12 | log ρ + ε|. At the end of the proof, we use Proposition 2 to estimate ν p,E .  

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We can finally deduce that, with probability exponentially close to 1, the matrix elements ((∧ pU (n) (E))x, y), for p ∈ {1, . . . , N} and x, y ∈ L p , grow exponentially at almost the rate of (γ1 + · · · + γ p )(E). Proposition 3 Let p ∈ {1, . . . , N} and assume that E ∈ R is such that G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Let x, y ∈ L p , x = y = 1. Then, for every ε > 0, there exist κ > 0 and n0 ∈ N such that,    ∀n ≥ n0 , P ( ∧ p U (n) (E) x, y) ≥ e(γ1 (E)+···+γ p (E)−ε)n ≥ 1 − e−κn .

(35)

Proof Let ε > 0 and x, y ∈ L p , x = y = 1. First, from Lemma 4, we deduce that there exists n1 ∈ N such that, for every n ≥ n1 ,    P e(γ1 (E)+···+γ p (E)−ε)n ≤ (∧ pU (n) (E))x ≤ e(γ1 (E)+···+γ p (E)+ε)n ≥ 1 − e−κ0 n . (36) Then, combining this probability estimate with (34), we get the existence of n2 ∈ N such that, for every n ≥ n2 ,     P (∧ pU (n) (E)x, y) ≥ e−εn  ∧ p U (n) (E)x ≥ e(γ1 (E)+···+γ p (E)−2ε)n ≥ 1 − e−κ1 n − e−κ0 n . We get (35) for n large enough, say n ≥ n0 , with κ = min(κ0 , κ1 ) > 0.

 

This result will be used in Section 6.2 to prove a probability estimate required to start a multiscale analysis.

4 The Integrated Density of States To prove a Wegner estimate for H(ω) like in Theorem 5, a crucial property is the local Hölder continuity of the integrated density of states of H(ω). We review, in this Section, the definition of the integrated density of states and we prove Theorem 4. The integrated density of states is the distribution function of the proper energy levels, per unit volume, of H(ω). To define it, we consider, for every integer L ≥ 1, the restriction H (L) (ω) of H(ω) to L2 ([−L, L]) ⊗ C N , with Dirichlet boundary conditions at ±L. Definition 5 The integrated density of states associated to H(ω) is the function from R to R+ , E → N(E), where N(E), for E ∈ R, is defined as the following thermodynamical limit:    1  # λ ≤ E λ ∈ σ H (L) (ω) , L→+∞ 2L

N(E) = lim for P-almost every ω ∈ .

(37)

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In this definition, appear two problems of existence. The first one is to verify that the cardinal in (37) is finite for every ω ∈ . The second one is the existence of the limit and its almost-sure independence on ω. In [3], we already proved the existence of the integrated density of states for matrixvalued continuous Schrödinger operators of the form (1). In particular, the integrated densities of states of H(ω) and H (ω) are well defined for every  > 0 and every E ∈ R. Moreover, as N(E) and the sum of positive Lyapunov exponents, γ1 (E) + · · · + γ N (E), are harmonically conjugated through a Thouless formula (see [3, Theorem 3]), N(E) inherits the same Hölder regularity as the Lyapunov exponents. This is how we proved [3, Theorem 4]. Applying this theorem to H (ω) and using Theorem 3, we obtain Theorem 4 as stated in the introduction. The Hölder exponent α in (9) is equal to the Hölder exponent of the Lyapunov exponents in (8). This is due to properties of the Hilbert transform.

5 A Wegner Estimate This Section is devoted to the proof of Theorem 5. For this purpose, we need two lemmas which give estimates on the norm of the solutions of the equation −u + Vu = 0 for V ∈ L1loc (R, MN (R)). Lemma 6 Let V be a matrix-valued function in L1loc (R, MN (R)) and u a solution of −u + Vu = 0. Then, for every x, y ∈ R,  2   2 u(x) + u (x) ≤ u(y)2 + u (y) exp



max(x,y)

2

 V(t) + 1 dt .

min(x,y)

(38) We already proved this lemma in [3, Lemma 2]. Lemma 7 Let V be a matrix-valued function in L1loc (R, MN (R)) such that, for * x+ a fixed  > 0, V,u = supx∈R x V(t)dt < ∞. Then there exists C > 0 such that, for every solution u of −u + Vu = 0 and every x ∈ R, 

x+

 u(t)2 dt ≥ C u(x)2 + u (x)2 .

(39)

x−

Proof Let x ∈ R and u be a solution of −u + Vu = 0. Applying Lemma 6 to x and t ∈ [x − , x + ], one gets:  2  2 u(t)2 + u (t) ≤ (u(x)2 + u (x) ) exp



max(t,x) min(t,x)

 V(s) + 1 ds

 2 ≤ u(x)2 + u (x) exp(2 + 2V,u ), 

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and  2  u(x) + u (x)| ≤ u(t)2 + u (t)2 exp



max(x,t)

2

 V(s) + 1 ds

min(x,t) 

≤ (u(t) + u (t) ) exp(2 + 2V,u ). 2

2

Setting C1 = exp(−2 − 2V,u ) and C2 = exp(2 + 2V,u ), we obtain, for every t ∈ [x − , x + ],    2   2  2  C1 u(x)2 + u (x) ≤ u(t)2 + u (t) ≤ C2 u(x)2 + u (x) . (40) 

We set Nx = u(x) + u (x). Using the inequality (a + b ) ≤ 2(a + b 2 ) valid for every a, b ∈ R, we have, for every t ∈ [x − , x + ], 2

2

   2   2  C1 Nx2 ≤ 2C1 u(x)2 + u (x) ≤ 2 u(t)2 + u (t) ≤ 2Nt2 ,

(41)

   2   2  Nt2 ≤ 2 u(t)2 + u (t) ≤ 2C2 u(x)2 + u (x) ≤ 2C2 Nx2 .

(42)

and

Setting C3 =



C1 2

 12

1

and C4 = (2C2 ) 2 , we obtain, for every t ∈ [x − , x + ],

        C3 u(x) + u (x) ≤ u(t) + u (t) ≤ C4 u(x) + u (x) ,

(43)

which is, with our notation, C3 Nx ≤ Nt ≤ C4 Nx , for every t ∈ [x − , x + ]. Assume that, for any t ∈ [x − , x + ], u(t) < C23 Nx . Then, u (t) > C23 Nx or else it would contradict (43). In particular, u does not vanish on [x − , x + ] and the signs of its coordinates remain constant on this interval. Thus, for t = x −  and t = x + , C3 N x =

C3 C3 Nx + Nx > u(x + ) + u(x − ) ≥ u(x + ) − u(x − ) 2 2  x+    u (s)ds > 2 C3 Nx = C3 Nx . = 2 x−

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We get a contradiction and thus, there exists t0 ∈ [x − , x + ], u(t0 ) ≥ Nx . But we also have, for every t ∈ [x − , x + ], u (t) ≤ C4 Nx . Let t ∈ C3 [x − , x + ] be such that |t − t0 | ≤ 4C . Then, we have: 4   u(t) = u(t0 ) + u(t) − u(t0 ) ≥ u(t0 ) − u(t) − u(t0 )   t       = u(t0 ) −  u (s)ds   C3 2

t0

C3 C3 C3 Nx − Nx = Nx , ≥ 2 4 4 because  t   t   t C3 C3   Nx . u (s)ds ≤ u (s)ds ≤ C4 Nx ds ≤ C4 N x ≤  4C4 4 t0 t0 t0 So, just proved that there exists an interval I0 of length  we have  C3 min 2, 2C4 , included in [x − , x + ], such that u(t) ≥ C43 Nx , for every t ∈ I0 . Then, 

x+

 u(t)2 dt ≥

x−

u(t)2 dt ≥ I0

  C32 C3 Nx2 min 2, 16 2C4

≥ C(u(x)2 + u (x)2 ). It proves the lemma.

 

We can now prove the following proposition upon which will be based the proof of Theorem 5. ˜ Proposition 4 Let I ⊂ R be a compact interval and I˜ be an open interval, I ⊂ I, ˜ G(E) is p-contracting and Lp -strongly irreducible, such that, for every E ∈ I, for every p ∈ {1, . . . , N}. Then, there exist α > 0, L0 ∈ N and C > 0 such that, for every E ∈ I and every ε > 0:    ∀L ≥ L0 , P ∃E ∈ (E − ε, E + ε), ∃φ ∈ D H (L) (ω)  H (L) (ω) − E φ = 0, +  2  2 φ = 1 and φ  (−L) + φ  (L) ≤ ε2 ≤ C  L εα . (44) Proof The proof will mostly relies on the Hölder continuity of the integrated ˜ density of states of H(ω). Let I˜ be an open interval such that, for every E ∈ I, G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Let I ⊂ I˜ be a compact interval. By [3, Theorem 3], there exist α > 0 and C1 > 0 such that:   α   ∀E, E ∈ I,  N(E) − N E  ≤ C1  E − E  . (45)

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Let E be in the interior of I, ε > 0 and L ∈ N. For every k ∈ Z, let Ik be the interval 2kL + [−L, L] = [(2k − 1)L, (2k + 1)L] and denote by H (Ik ) (ω) the restriction of H(ω) to L2 (Ik ) ⊗ C N with Dirichlet boundary conditions. We define the event Ak ∈ A as: Ak = {ω ∈  | H (Ik ) (ω) has an eigenvalue λk ∈ (E − ε, E + ε) such that the corresponding normalized eigenfunction φk satisfies φ  (−L)2 + φ  (L)2 ≤ ε2 }. Then, because the Vω(n) are i.i.d. random variables, and because of the form of the potential in H(ω) as a -periodization in law of Vω(n) , we deduce that P(Ak ) is independent of k, that is, ∀k ∈ Z, P(Ak ) = P(A0 ). Moreover, P(A0 ) is equal to the probability in (44). Let n ∈ N and let Jn = ∪nk=−n Ik = [−(2n + 1)L, (2n + 1)L]. Let H (Jn ) (ω) be the restriction of H(ω) to L2 (Jn ) ⊗ C N with Dirichlet boundary conditions. For a fixed ω ∈ , let k1 , . . . , k j ∈ {−n, . . . , n} be distinct and such that ω ∈ Aki for every i ∈ {1, . . . , j}. Let i ∈ {1, . . . , j}. Let φi be defined on Iki , φi  = 1, φi ((2ki − 1)L) = φi ((2ki + 1)L) = 0. We also assume that there exists λki ∈ (E − ε, E + ε) such that H (Iki ) (ω)φi = λki φi . Let χ be a smooth function on R, 0 ≤ χ ≤ 1, χ(x) = 0 on (−∞, 0], χ(x) = 1 * on [, +∞) and 0 χ(x)dx = 1. Let xi± = (2ki ± 1)L, so that Iki = [xi− , xi+ ]. We extend φi to Jn by defining φˆ i , for x ∈ Jn , by: ⎧ 0 if ⎪ ⎪ ⎨  − χ x − x φ (x) if i i φˆ i (x) = φi(x) if ⎪ ⎪ ⎩ χ xi+ − x φi (x) if

− + x∈ / [x  i−, xi−]  x ∈ xi , xi +   x ∈ xi− + , xi+ −  x ∈ xi+ − , xi+ .

(46)

Then, φˆ i ∈ D(H (Jn ) (ω)) and φˆ i  ≤ φi  = 1. As H (Iki ) (ω)φi = λki φi and λki ∈ (E − ε, E + ε), we have:  (J )       H n (ω) − E φˆ i  ≤  H (Jn ) (ω) − λk φˆ i  +  λk − E φˆ i  i i      =  H (Jn ) (ω) − λki φˆ i  + λki − E φˆ i    ≤  H (Jn ) (ω) − λki φˆ i  + ε.

(47)

We want to estimate (H (Jn ) (ω) − λki )φˆ i . For every x ∈ [xi− , xi− + ],       χ x − xi− φi (x) = χ  x − xi− φi (x) + 2χ  x − xi− φi (x) + χ x − xi− φi (x), and, for every x ∈ [xi+ − , xi+ ],   +     χ xi − x φi (x) = χ  xi+ − x φi (x) − 2χ  xi+ − x φi (x) + χ xi+ − x φi (x).

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Thus, using H (Iki ) (ω)φi = λki φi ,  (Jn ) H (ω) − λki φˆ i ⎧ 0  if ⎪ ⎪  ⎨ −χ  x − xi− φi (x) − 2χ  x − xi− φi (x) if = 0  if ⎪ ⎪  ⎩ −χ  xi+ − x φi (x) + 2χ  xi+ − x φi (x) if

  x∈ / xi− , xi+  x ∈ xi− , xi− +   x ∈ xi− + , xi+ −  x ∈ xi+ − , xi+ .

Hence we have, applying twice Lemma 6 for xi− and for xi+ at the second inequality,  (J )   H n (ω) − λk φˆ i 2 = i



xi− + xi−



+

     χ x − x− φi (x) + 2χ  x − x− φ  (x)2 dx i i i

xi+ xi+ −

   +   χ x − x φi (x) − 2χ  x+ − x φ  (x)2 dx i i i

 2   xi− +     χ x − x− 2  φi  i  dx    ≤ ×  2χ  x − x−   φ  ∞ − − i i x− L ([x ,x +]) i

i

i

 2   xi+    +  φi   χ x − x 2 i     × +  φ  ∞ +  2χ  x+ − x  dx i i x+ − L ([x −,x+ ]) i

i

i

  −    + 2  φi x     i   φi xi  ≤ C2   φ  x−  +  φ  x+  i i i i   − 2   + 2 = C2 φi xi  + φi xi  ≤ C2 ε2 ,

(48)

using the fact that ω ∈ Aki and using the Dirichlet boundary conditions of H (Iki ) (ω) at xi− and xi+ to say that φi (xi− ) = φi (xi+ ) = 0. The constant C2 depends only on χ and the parameters of the potential of H(ω). We normalize φˆ i by * setting φ˜ i = φˆ i /φˆ i . We also have φˆ i  ≥ 12 because φi  = 1 and 0 χ(x)dx = 1, and thus, by (47) and (48),  (J )       H n (ω) − E φ˜ i  = φˆ i −1  H (Jn ) (ω) − E φˆ i 

(49) ≤ 2 C2 ε := C3 ε. We have construct, for each i ∈ {1, . . . , j}, a normalized function φ˜ i in D(H (Jn ) (ω)), supported in Iki , such that:   ∀i ∈ {1, . . . , j },  H (Jn ) (ω) − E φ˜ i  ≤ C3 ε, (50) where C3 depends only on the choice of χ and on the parameters of the potential of H(ω). Moreover, as φ˜ i is supported in Iki and the intervals Ik1 , . . . , Ik j are disjoints, (φ˜ 1 , . . . , φ˜ j) is an orthonormal set and:   (51) ∀i = i , φ˜ i , H (Jn ) (ω)φ˜ i = 0 = H (Jn ) (ω)φ˜ i , H (Jn ) (ω)φ˜ i .

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We recall that, as proven in [3, Section 2.3], the spectrum of H (Jn ) (ω) is a discrete set of eigenvalues with only +∞ as accumulation point, and thus, its number of eigenvalues in any compact interval is finite. As we have (50) and (51), we can apply to (φ˜ 1 , . . . , φ˜ j) and H (Jn ) (ω) the version of Temple’s inequality given in [19, Lemma A.3.2] to obtain that the number of eigenvalues of H (Jn ) (ω) in [E − C3 ε, E + C3 ε], counted with multiplicity, is at least j. So we have, for a fixed ω ∈ ,        j= # k ∈ {−n, . . . , n}  ω ∈ Ak ≤ # λ ∈ [E − C3 ε, E + C3 ε]  λ ∈ σp H (Jn ) (ω) . Moreover, applying the law of large numbers to the random variables 1 A−n , . . . , 1 An , we get that, for P-almost every ω ∈ ,     1 1  # k ∈ {−n, . . . , n}  ω ∈ Ak = 1 A−n +. . . + 1 An −−−−→ E 1 A0 , n→+∞ 2n + 1 2n + 1 with E(1 A0 ) = P(A0 ). Now, we assume that ε is small enough to ensure that [E − C3 ε, E + C3 ε] ⊂ I ⊂ I˜ and to apply (45) on [E − C3 ε, E + C3 ε]. Then we have, for P-almost every ω ∈ ,    1 # k ∈ {−n, . . . , n}  ω ∈ Ak n→+∞ 2n + 1     1 # λ ∈ [E − C3 ε, E + C3 ε]  λ ∈ σp H (Jn ) (ω) ≤ lim n→+∞ 2n + 1     1 # λ ∈ [E − C3 ε, E + C3 ε]  λ ∈ σp H (Jn ) (ω) = 2L lim n→+∞ 2(2n + 1)L

 P A0 = lim

= 2L(N(E + C3 ε) − N(E − C3 ε)) ≤ 2LC1 (2C3 ε)α := CLεα . It finishes the proof.

 

We remark that the exponent α in (44) is the same as the Hölder exponent of the Lyapunov exponent and the integrated density of states. We can now use Proposition 4, Lemmas 3 and 4 to prove Theorem 5. Proof (of Theorem 5) Let I ⊂ R be a compact interval and I˜ be an open ˜ such that, for every E ∈ I, ˜ G(E) is p-contracting and Lp interval, I ⊂ I, strongly irreducible, for every p ∈ {1, . . . , N}. Let β ∈ (0, 1) and κ > 0. For L ∈ N, we set n L = [τ (L)β ] + 1 with some arbitrary τ > 0, where [τ (L)β ] is the largest integer less or equal to τ (L)β . For every E ∈ I and θ0 > 0, we define the events:       1  (L) θ0 (L)β  (n −L−1) (−L) Aθ0 (E) = ω ∈    , (52) T (E) . . . T (E) > e L ω  ω 0        0  (L) θ0 (L)β    Bθ0 (E) = ω ∈   Tω(L+1−nL ) (E) . . . Tω(L) (E) >e . (53) 1 

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H. Boumaza

Let ξ1 > 0 and δ > 0 be the constants given by Lemma 3. Let θ = C (E) ∈ A be the event:

τ ξ1 2δ

(L)

-

and let

  +  β  ω ∈   d E, σ H (L) (ω) ≤ e−κ(L) .

If we set: ⎛

.



(a) := P ⎝C(L) (E) ∩

(L) 

Aθ

2

β

⎛

{E | |E−E |≤e−κ(L) }

{E |

(L) 

E ∩ Bθ

2



0

(L) (b ) := P ⎝ A(L) θ (E) ∩ Bθ (E) ∩

⎛



(L) 

Aθ

(L) (c) := P ⎝ A(L) θ (E) ∩ Bθ (E) ∩



(L) 

Bθ β

(d) := P



E

 c 

2

β |E−E |≤e−κ(L) }

0

⎞ / E ⎠ ,

{E | |E−E |≤e−κ(L) }

E

c 

2

⎞ ⎠,

⎞ ⎠,

c  c   + P B(L) , A(L) θ (E) θ (E)

then we have: P

-

  +  β  ≤ (a) + (b ) + (c) + (d). ω ∈   d E, σ H (L) (ω) ≤ e−κ(L)

(54)

Using Tchebychev’s inequality and Lemma 3, applied for p = 1, we directly get, for L large enough, β

β

(d) ≤ 2e−ξ1 nL −δθ(L) ≤ 2e−ξ1 τ (L)

+

τ ξ1 2

(L)β

= 2e−

τ ξ1 2

(L)β

.

(55)

To estimate (b ) + (c), we use the fact that there exists a constant C0 > 0 independent of n, ω, E such that, for every E, E ∈ I,      ∀n ∈ Z, Tω(n) (E) − Tω(n) E  ≤ C0  E − E .

(56)

It wasproven in the proof of [3, Theorem 2]. From this, we deduce that the c   event A(L) (E ) ∩ A(L) θ θ (E) occurs for at least one E such that |E − E | ≤ β

2

e−κ(L) . Then, following [8], we prove that for this E , there exists α0 > 0 such that, if τ > 0 is small enough,   c  β (L)   (E) ∩ A P A(L) E ≤ e−α0 (L) . θ θ 2

(57)

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279

 c (L)  We have a similar inequality for B(L) (E) ∩ B (E ) for at least one E θ θ 2

β

such that |E − E | ≤ e−κ(L) . Thus, by inclusions of the events:     c  c  (L)   (L)   + P B(L) (b ) + (c) ≤ P A(L) E E θ (E) ∩ A θ θ (E) ∩ B θ 2

2

β

≤ 2e−α0 (L) .

(58)

It remains to estimate (a). Let ω be in the event in the probability (a). Let β β E ∈ (E − e−κ(L) , E + e−κ(L) ) be an eigenvalue of H (L) (ω) with a normalized eigenvector φ. As ω is in the event in the probability (a), we have, using Lemma 4,  2  2 β φ(−L)2 + φ  (−L) = φ  (−L) ≤ 2e−θ (L) ,

(59)

 2  2 β φ(L)2 + φ  (L) = φ  (L) ≤ 2e−θ (L) .

(60)

and

β

Now, using Proposition 4 with ε = e−κ(L) , we get:   √ θ β β α (a) ≤ CL max e−κ(L) , 2 2e− 2 (L) .

(61)

Putting (55), (58) and (61) in (54), we finally obtain (10) for a suitable ξ > 0 and L large enough.  

6 Localization Properties for H(ω) and H (ω) In this section, we will prove Theorem 1 and its corollary, Theorem 2. It will be the content of Section 6.2. Before that, we will present in Section 6.1, the requirements needed to perform a multiscale analysis. 6.1 Requirements of the Multiscale Analysis In this Section we present the properties of H(ω) needed to use the multiscale analysis. These properties are, for most of them, already detailed in [20] and [11], but we will follow here the notations of [16], based upon [12], as we did in the introduction for the definitions of spectral and dynamical localization. We start by giving a property that guarantees the existence of a generalized eigenfunction expansion for H(ω). If we denote by H the Hilbert space L2 (R) ⊗ C N , given ν > 14 , we define the weighted spaces H± by:  H± = L2 R, < x >±4ν dx ⊗ C N ,

280

H. Boumaza

where < x > is as in the introduction, equal to 1 + |x|2 , for any x ∈ R. We define on H+ × H− the sesquilinear form < , >H+ ,H− by:  t ∀(φ, ψ) ∈ H+ × H− , < φ, ψ >H+ ,H− = φ(x)ψ(x)dx. R

We also set T to be the self-adjoint operator on H given by the multiplication by < x >2ν . We recall that Eω (.) denotes the spectral projection of H(ω) and we present a property of Strong Generalized Eigenfunction Expansion. Definition 6 Let I ⊂ R be an open interval. We say that H(ω) has the property (SGEE) on I if, for some ν > 14 , (i) for P-almost every ω ∈ , the set D+ (ω) = {φ ∈ D(H(ω)) ∩ H+ | H(ω)φ ∈ H+ } is dense in H+ and is an operator core for H(ω), (ii) there exists a bounded, continuous function f on R, strictly positive on σ (H(ω)) such that:   2  E trH T −1 f (H(ω))Eω (I)T −1 < ∞. Now, we can give the definition of a generalized eigenfunction and of a generalized eigenvalue. Definition 7 A measurable function ψ : R → C N is said to be a generalized eigenfunction of H(ω) with generalized eigenvalue λ if ψ ∈ H− \ {0} and: ∀φ ∈ D+ (ω), < H(ω)φ, ψ >H+ ,H− = λ < φ, ψ >H+ ,H− . We now introduce definitions and notations for the restrictions of H(ω) to intervals of R of finite length. For x ∈ Z and L ≥ 1, we denote by I L (x) the interval I L (x) = [x − L, x + L], centered at x and of length 2L. As in the introduction, we denote by 1x,L the characteristic function of I L (x) and simply by 1x , the characteristic function of I1 (x). For L ∈ 3N∗ , we also set, 1out x,L = 1x,L − 1x,L−2

and

1in x,L = 1x, L . 3

(x,L)

For every x ∈ Z and every L ≥ 1, we denote by H (ω) the restriction of H(ω) to L2 (I L (x)) ⊗ C N with Dirichlet boundary conditions, and, for E∈ / σ (H (x,L) (ω)), by R(x,L) (E) the resolvent of H (x,L) (ω) at E, R(x,L) (E) = (H (x,L) (ω) − E)−1 . We also denote by Eω(x,L) the spectral projection of H (x,L) (ω). With all these notations, we can state the following Simon-Lieb type inequality property. Definition 8 Let I ⊂ R be a compact interval. We say that H(ω) has the property (SLI) if there exists a constant C I such that, given L, L , L ∈ N and x, y, y ∈ Z, with I L (y) ⊂ I L −2 (y ) ⊂ I L−2 (x), for P-almost every ω ∈ ,   if E ∈ I, E ∈ / σ (H (x,L) (ω)) ∪ σ (H (y ,L ) (ω)), we have:  out (x,L)     (y ,L ) (x,L) 1 R  (E)1 y,L  ≤ C I 1out (E)1 y,L  1out (E)1out x,L y ,L R x,L R y ,L .

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281

The property (SLI) is an estimate of how the finite length resolvents R(x,L) (E) vary in norms when we go from one interval to a larger one containing the first one. It is also called a Geometric Resolvent Inequality in [20]. We now state a property which is an estimate of generalized eigenfunctions in terms of finite length resolvents. It is called an Eigenfunction Decay Inequality. Definition 9 Let I ⊂ R be a compact interval. We say that H(ω) has the property (EDI) if there exists a constant C˜ I such that, for P-almost every ω ∈ , given a generalized eigenvalue E ∈ I, we have for any x ∈ Z and any L ∈ N with E ∈ / σ (H (x,L) (ω)),      1x ψ  ≤ C˜ I 1out R(x,L) (E)1x  1out ψ . x,L x,L The next property is an estimate of the average number of eigenvalues of H (x,L) (ω). Definition 10 Let I ⊂ R be a compact interval. We say that H(ω) has the property (NE) if there exists a finite constant Cˆ I such that, for every x ∈ Z and L ∈ N,   E trH Eω(x,L) (I) ≤ Cˆ I L. The last property required for the multiscale analysis is of a different nature. It is a probabilistic property of independence of distant intervals. An event A ∈ A is said to be based on I L (x) if it is determined by conditions on H (x,L) (ω). Given d0 > 0, we say that I L (x) and I L (x ) are d0 -nonoverlapping if d(I L (x), I L (x )) > d0 . Definition 11 We say that H(ω) has the property (IAD) if there exists d0 > 0 such that events based on d0 -nonoverlapping intervals are independent. Before giving the definition of the multiscale analysis set MSA , we need a last definition. Definition 12 Let γ , E ∈ R and ω ∈ . For x ∈ Z and L ∈ 3N∗ , we say that the / σ (H (x,L) (ω)) and interval I L (x) is (ω, γ , E)-good if E ∈  out (x,L)  −γ  L3 1 R  (E)1in . x,L x,L ≤ e We can now define the multiscale analysis set. We assume that H(ω) has the property (IAD).

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H. Boumaza

Definition 13 The set MSA for H(ω) is the set of E ∈  for which there exists an open interval I such that E ∈ I and, given any ζ , 0 < ζ < 1, and α0 ∈ (1, ζ −1 ), there is a length scale L0 ∈ 6N and a real number γ > 0, so if we set Lk+1 = max{L ∈ 6N | L ≤ Lαk0 } for every k ∈ N, we have: P

  ζ ω ∈  | ∀E ∈ I, I L (x) or I L (y) is (ω, γ , E ) − good ≥ 1 − e−Lk .

for every k ∈ N and x, y ∈ Z with |x − y| > Lk + d0 . We finish this Section by stating the bootstrap multiscale analysis theorem of [12, Theorem 3.4] for operators involving singular probability measure like H(ω). Theorem 7 ([12], Theorem 3.4) Assume that H(ω) has the properties (IAD), (SLI), (NE) and verify a Wegner estimate (W) like (10) on an open interval I ⊂ R. Given γ > 0, for each E ∈ I, there exists an integer Lγ (E), bounded on compact subintervals of I, such that, if for a given E0 ∈  ∩ I we have: P

  ω ∈  | I L0 (0) is (ω, γ , E0 ) − good ≥ 1 − e−δL ,

(62)

for L0 ∈ N, L0 > Lγ (E) and δ > 0, then E0 ∈ MSA . The assumption (62) is also known as an Initial Length Scale Estimate (ILSE) in [10] and essentially, it remains to prove such an (ILSE) for our operator H(ω) on a valid interval, to prove localization on this interval. It is the main purpose of the next section. 6.2 Proof of the Localization for H(ω) and H (ω) To prove Theorems 1 and 2, we have to establish a link between multiscale analysis and the properties (EL), (SDL) and (SSEHSKD) defined in the introduction. This link is established in the following theorem. Theorem 8 ([16], Theorem 6.1) Let I ⊂ R be an open interval on which H(ω) has the properties (IAD), (SGEE) and (EDI). Then: MSA ∩ I ⊂ EL ∩ SSEHSKD ∩ I ⊂ EL ∩ SDL ∩ I. According to this theorem, to prove Theorems 1 and 2, it only remains to prove an (ILSE) for H(ω) to be able to apply Theorem 7 for every energies on

Localization for a Matrix-valued Anderson Model

283

a suitable interval. We can summarize in the following figure, the ingredients of a proof of localization using multiscale analysis. (IAD) + (SLI) + (NE) + (W) + (ILSE) 1 23 4 ⇓

(MSA) + (SGEE) + (EDI) 1 23 4 ⇓

3 41 2 (EL) + (SDL) + (SSEHSKD)

(63)

Proposition 5 Let I ⊂ R be an open interval such that, for every E ∈ I, G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. Let E ∈ I. For every ε > 0, there exist δ > 0 and L0 ∈ N such that, for every L ≥ L0 , L ∈ 3N∗ ,   P I L (0) is (ω, γ1 (E) − ε, E) − good ≥ 1 − e−δL . (64) Proof We fix E ∈ I and assume that L ∈ 3N∗ . We consider U + and U − two matrices in MN (R), solutions of H(ω)U ± = EU ± and such that: U + (L) = U − (−L) = 0





and U + (L) = U − (−L) = IN .

(65)

Let W(U + , U − ) denote the matrix-valued Wronskian of U + and U − defined by: 



∀x ∈ I L (0), W(U + , U − )(x) = t U − (x)U + (x) − t U − (x)U + (x).

(66)

From [9, Proposition III.5.5], W(U + , U − ) is constant on I L (0) and it is non-invertible if and only if E is an eigenvalue of H (L) (ω). We recall that the spectrum of H (L) (ω) consists on a discrete set of eigenvalues of H (L) (ω) with only +∞ as accumulation point. Thus, E ∈ σ (H (L) (ω)) if and only if W(U + , U − ) is non-invertible in MN (R). By [9, Proposition III.5.6], the Green kernel of H (L) (ω) is given by: ∀E ∈ / σ (H (L) (ω)), G(L) (E, x, y) =



U − (x)W(U − , U + )−1 t U + (y) if x ≤ y U + (x)W(U + , U − )−1 t U − (y) if x > y. (67)

To estimate the norm of R(0,L) (E), we can estimate the norm of its kernel, the Green kernel G(L) (E, x, y). We start by estimating the Wronskian. As it is constant, we have, using (66), W(U + , U − ) = W(U + , U − )(L) = −t U − (L),

(68)

and W(U + , U − ) = U − (L). But, if X, Y ∈ M2N×N (R), applying Proposition 3 for p = 1, column by column, we have:  ∃n0 ≥ 1, ∀n ≥ n0 , P t YU (n) (E)X ≥ e(γ1 (E)−ε)n ≥ 1 − e−κn , (69)

284

H. Boumaza

L with ε > 0 and κ > 0 as in the proposition. If T−L (E) denote the transfer L matrix from −L to L, we have U − (L) = (IN , 0)T−L (E)t (IN , 0). Thus, t t applying (69) for Y = (IN , 0) and X = (IN , 0), and using the fact that the transfer matrices are i.i.d.,

 ∃L1 ≥ 1, ∀L ≥ L1 , P W(U + , U − ) ≥ e2(γ1 (E)−ε)L ≥ 1 − e−2κL .

(70)

/ Let x ∈ [L − , L] and y ∈ [− L3 ,  L3 ]. Then x > y and for E ∈ (L) (L) −1 t σ (H (ω)), G (E, x, y) = U + (x)W(U + , U − ) U − (y). We apply Lemma 6 to U + , for x and y = L, and using (65): U + (x) ≤ C1 ,

(71)

with C1 independent of ω and L. To estimate the norm of t U − (y) is more complicated. We start by writing:       U − (y)   −L 0          U − (y) ≤  = T y (E)    I U − (y) N     −L 0  ≤ T y[y+]− (E) T[y+]− (E) . IN  t

By Lemma 6 for y and −L and using (65), we have T y[y+]− (E) ≤ C2 ,

(72)

with C2 independent of ω and L. Now, using the i.i.d. character of the transfer matrices and (36) for p = 1, we get the existence of L2 ∈ N such that,      −L L 0  (γ1 (E)+ε)4 L3 (E) ≥ e ≥ 1 − e−2κ0  3 . ∀L ≥ 3L2 , P T[y+]− IN 

(73)

If C = max(C1 , C2 ), using (70), (71), (72) and (73), we get, for L ≥ max(L1 , 3L3 ),   L P E∈ / σ (H (L) (ω)) and G(L) (E, x, y) ≤ Ce−(γ1 (E)−10ε) 3 ≥ 1 − e2κL − e−2κ0  3 . L

(74)

Now, if we assume that x ∈ [−L, −L + ] and y ∈ [− L3 ,  L3 ], we have x ≤ y and for E ∈ / σ (H (L) (ω)), G(L) (E, x, y) = U − (x)W(U − , U + )−1 t U + (y). In a similar way as we proved (74), we get the same estimate:   L P E∈ / σ (H (L) (ω)) and G(L) (E, x, y) ≤ Ce−(γ1 (E)−10ε) 3 ≥ 1 − e2κL − e−2κ0  3 . L

(75)

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We introduce the events A L,ε (E) and B L,ε (E) defined by:   A L,ε (E) = ω ∈   E ∈ / σ (H (L) (ω)) and  + (L) in −(γ1 (E)−ε) L3 sup 1out (x)G (E, x, y)1 (y)dy ≤ e 0,L 0,L x∈R

and

R

  B L,ε (E) = ω ∈   E ∈ / σ (H (L) (ω)) and  + (L) −(γ1 (E)−ε) L3 sup 1out (E, x, y)1in . 0,L (x)G 0,L (y)dx ≤ e y∈R

R

Then, from (74) and (75), we deduce that, for every ε > 0, there exist δ > 0 and L0 ∈ N, L0 ≥ max(L1 , 3L2 ), such that,  ∀L ≥ L0 , L ∈ 3N, P A L,ε (E) ∩ B L,ε (E) ≥ 1 − e−δL . To pass from this estimate on the kernel G(L) (E, x, y) of R(0,L) (E) to the estimate (64) on R(0,L) (E), we use Schur’s test. It finishes the proof.   It is interesting here to remark that the exponential decaying rate of the resolvent, and thus of the eigenfunctions of H(ω), is almost γ1 (E), the biggest positive Lyapunov exponent times the interaction length . We have now all the requirements needed to prove Theorems 1 and 2. Proof (of Theorem 1) Let I ⊂ R be a compact interval,  ∩ I = ∅, and let I˜ be ˜ such that, for every E ∈ I, ˜ G(E) is p-contracting and an open interval, I ⊂ I, Lp -strongly irreducible, for every p ∈ {1, . . . , N}. If we look at the proof of [13, Theorem A.1], we see that the potential only appears through estimates of its absolute value and so, changing the absolute value into a matrix-norm in this proof, we get that H(ω) has the properties (SLI), (EDI), (NE) and (SGEE) on I. From the form of the potential of H(ω) and the assumption on independence of the Vω(n) , H(ω) also has the property (IAD). By Theorem 5, H(ω) verifies a Wegner estimate (W) on I, and by Proposition 5, it verifies an (ILSE) estimate on I. So we can apply Theorem 7 for every E0 ∈ I to get I ⊂ MSA . Then, applying Theorem 8, we get that: I ⊂ EL ∩ SSEHSKD ⊂ EL ∩ SDL . It proves Theorem 1.

 

Proof (of Theorem 2) For the point (i), by Proposition 1, for  < C and for every E ∈ I(, N), G(E) is p-contracting and Lp -strongly irreducible, for every p ∈ {1, . . . , N}. As I(, N) is a compact interval, we can apply Theorem 1 to H (ω) on every open interval I ⊂ I(, N) such that  ∩ I = ∅. We already remark in the introduction that such open intervals exist. For the point (ii), we use [2, Proposition 2] for the existence of a discrete set S ⊂ R such that for every E ∈ (2, +∞) \ S , G(E) is p-contracting and Lp -

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strongly irreducible, for every p ∈ {1, . . . , N}. Once again, if I ⊂ (2, +∞) \ S is a compact interval, as S is discrete, there always exists an open interval I˜ ⊂ ˜ Therefore, we can apply Theorem 1 to H1 (ω) for N = 2 on (2, +∞) \ S , I ⊂ I. I ⊂ (2, +∞) \ S , I compact and such that  ∩ I = ∅.   Acknowledgements The author is supported by JSPS Grant P07728. This paper was written during his post-doctoral stay at Keio University in Yokohama. He would like to thank Yoshiaki Maeda to allow him the opportunity of working in excellent conditions at the Keio mathematics department. He also would like to thank the referee for numerous remarks which helped him to improve the introduction of this paper. Finally, the author would like to thank Anne Boutet de Monvel for her constant encouragements during this work.

References 1. Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Progr. Probab. Statist., vol. 8. Birkhäuser, Boston (1985) 2. Boumaza, H.: Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model. Math. Phys. Anal. Geom. 10(2), 97–122 (2007). doi:10.1007/s11040-007-9023-6 3. Boumaza, H.: Hölder continuity of the integrated density of states for matrix-valued Anderson models. Rev. Math. Phys. 20(7), 873–900 (2008). doi:10.1142/S0129055X08003456 4. Boumaza, H.: Lyapunov exponents and integrated density of states for matrix-valued continuous Schrödinger operators. Thèse de l’Université Denis Diderot-Paris 7. http://tel. archives-ouvertes.fr/tel-00264341/fr/ (2007) 5. Boumaza, H., Stolz, G.: Positivity of Lyapunov exponents for Anderson-type models on two coupled strings. Electron. J. Differential Equations 47, 1–18 (2007) 6. Bourgain, J., Kenig, C.E.: On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math. 161(2) 389–426 (2005) 7. Breuillard, E., Gelander, T.: On dense free subgroups of Lie groups. J. Algebra 261(2), 448– 467 (2003) 8. Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Comm. Math. Phys. 108, 41–66 (1987) 9. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Probability and Its Applications. Birkhäuser, Boston (1990) 10. Damanik, D., Sims, R., Stolz, G.: Localization for one-dimensional, continuum, BernoulliAnderson models. Duke Math. J. 114, 59–99 (2002) 11. Damanik, D., Stollmann, P.: Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11(1), 11–29 (2001) 12. Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Comm. Math. Phys. 222(2), 415–448 (2001) 13. Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124(2), 309–350 (2004) 14. Gol’dsheid, I.Y., Margulis, G.A.: Lyapunov indices of a product of random matrices. Russian Math. Surveys 44(5), 11–71 (1989) 15. Kirsch, W., Molchanov, S., Pastur, L., Vainberg, B.: Quasi 1D localization: deterministic and random potentials. Markov Process. Related Fields 9, 687–708 (2003) 16. Klein, A.: Multiscale analysis and localization of random operators. arXiv:0708.2292v1 (2007) 17. Klein, A., Lacroix, J., Speis, A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94, 135–155 (1990) 18. Kotani, S., Simon, B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119(3), 403–429 (1988) 19. Simon, B., Taylor, M.: Harmonic analysis on SL(2, R) and smoothness of the density of states in the one-dimensional Anderson model. Comm. Math. Phys. 101(1), 1–19 (1985) 20. Stollmann, P.: Caught by Disorder—Bound States in Random Media. Progress in Mathematical Physics, vol. 20. Birkhäuser, Boston (2001)

Math Phys Anal Geom (2009) 12:287–324 DOI 10.1007/s11040-009-9062-2

Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent Katrin Grunert · Gerald Teschl

Received: 3 August 2008 / Accepted: 19 June 2009 / Published online: 3 July 2009 © Springer Science + Business Media B.V. 2009

Abstract We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method. Keywords Riemann–Hilbert problem · KdV equation · Solitons Mathematics Subject Classifications (2000) Primary 37K40 · 35Q53 · Secondary 37K45 · 35Q15 1 Introduction One of the most famous examples of completely integrable wave equations is the Korteweg–de Vries (KdV) equation qt (x, t) = 6q(x, t)qx (x, t) − qxxx (x, t),

(x, t) ∈ R × R,

Research supported by the Austrian Science Fund (FWF) under grant no. Y330. K. Grunert · G. Teschl (B) Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria e-mail: [email protected] URL: http://www.mat.univie.ac.at/∼gerald/ K. Grunert e-mail: [email protected] URL: http://www.mat.univie.ac.at/∼grunert/ G. Teschl International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

(1.1)

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where, as usual, the subscripts denote the differentiation with respect to the corresponding variables. Following the seminal work of Gardner et al. [17], one can use the inverse scattering transform to establish existence and uniqueness of (real-valued) classical solutions for the corresponding initial value problem with rapidly decaying initial conditions. We refer to, for instance, the monographs by Marchenko [27] or Eckhaus and Van Harten [16]. Our concern here are the long-time asymptotics of such solutions. The classical result is that an arbitrary short-range solution of the above type will eventually split into a number of solitons travelling to the right plus a decaying radiation part travelling to the left, as illustrated in Fig. 1. The first numerical evidence for such a behaviour was found by Zabusky and Kruskal [39]. The first mathematical results were given by Ablowitz and Newell [1], Manakov [26], and Šabat [30]. First rigorous results for the KdV equation were proved by Šabat [30] and Tanaka [34] (see also Eckhaus and Schuur [15], where more detailed error bounds are given). Precise asymptotics for the radiation part were first formally derived by Zakharov and Manakov [38], by Ablowitz and Segur [2, 32], by Buslaev [6] (see also [5]), and later on rigorously justified and extended to all orders by Buslaev and Sukhanov [7]. A detailed rigorous proof (not requiring any a priori information on the asymptotic form of the solution) was given by Deift and Zhou [10] based on earlier work of Manakov [26] and Its [19] (see also [20–22]). For further information on the history of this problem we refer to the survey by Deift et al. [12]. To describe the asymptotics in more detail, we recall the well-known fact (see e.g. [9, 27]) that q(x, t) is uniquely determined by the (right) scattering data of the associated Schrödinger operator H(t) = −

d2 + q(x, t). dx2

(1.2)

The scattering data consist of the (right) reflection coefficient R(k, t), a finite number of (t independent) eigenvalues −κj2 with 0 < κ1 < κ2 < · · · < κ N , and

Fig. 1 Numerically computed solution q(x, t) of the KdV equation at time t = 5, with initial condition q(x, 0) = sech(x + 3) − 5 sech(x − 1)

100

− 50

50

−2

−4

−6

100

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norming constants γj(t). We will write R(k) = R(k, 0) and γj = γj(0) for the scattering data of the initial condition. Then the long-time asymptotics can be described by distinguishing the following main regions: (i) The soliton region, x/t > C for some C > 0, in which the solution is asymptotically given by a sum of one-soliton solutions q(x, t) ∼ −2

N  j=1



κj2

cosh2 κj x − 4κj3 t − pj

,

(1.3)

where the phase shifts are given by ⎞ ⎛ N  γj2  κl − κj 2 1 ⎠. pj = log ⎝ 2 2κj κl + κj

(1.4)

l= j+1

In the case of a pure N-soliton solution (i.e., R(k, t) = 0) this was first established independently by Hirota [18], Tanaka [33], and Wadati and Toda [36]. The general case was first established by Šabat [30] and by Tanaka [34] (see also [15] and [31]). (ii) The self-similar region, |x/(3t)1/3 |  C for some C > 0, in which the solution is connected with the Painléve II transcendent. This was first established by Segur and Ablowitz [32]. −x (iii) The collisionless shock region, x < 0 and C−1 < (3t)1/3 (log(t)) 2/3 < C, for some C > 1, which only occurs in the generic case (i.e., when R(0) = −1). Here the asymptotics can be given in terms of elliptic functions as was pointed out by Segur and Ablowitz [32] with further extensions in Deift et al. [13]. (iv) The similarity region, x/t < −C for some C > 0, where 

4ν(k0 )k0 1/2 sin 16tk30 − ν(k0 ) log 192tk30 + δ(k0 ) , (1.5) q(x, t) ∼ 3t with ν(k0 ) = −

  1 log 1 − |R(k0 )|2 , 2π

 N  κj π arctan δ(k0 ) = − arg(R(k0 )) + arg((iν(k0 ))) + 4 4 k 0 j=1 +

1 π



k0

−k0



log(|ζ − k0 |)d log 1 − |R(ζ )|2 .

 x Here k0 = − 12t denotes the stationary phase point, R(k) = R(k, t = 0) the reflection coefficient, and  the Gamma function.

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Again this was found by Zakharov and Manakov [38] and (without a precise expression for δ(k0 ) and assuming absence of solitons) by Ablowitz and Segur [2] with further extensions by Buslaev and Sukhanov [7] as discussed before. Our aim here is to use the nonlinear steepest descent method for oscillatory Riemann–Hilbert problems from Deift and Zhou [10] and apply it to rigorously establish the long-time asymptotics in the soliton and similarity regions (Theorem 4.4, respectively, 5.4, below). In fact, our main goal is to give a complete and expository introduction to this method. In addition to providing a streamlined and simplified approach, the following items will be different in comparison with [10]. First of all, in the mKdV case considered in [10] there were no solitons present. We will add them using the ideas from Deift et al. [14] following Krüger and Teschl [24]. However, in the presence of solitons there is a subtle nonuniqueness issue for the involved Riemann–Hilbert problems (see e.g. [4, Chap. 38]). We will rectify this by imposing an additional symmetry condition and prove that this indeed restores uniqueness. Secondly, in the mKdV case the reflection coefficient R(k) has always modulus strictly less than one. In the KdV case this is generically not true and hence terms of the form R(k)/(1 − |R(k)|2 ) will become singular and cannot be approximated by analytic functions in the sup norm. We will show how to avoid these terms by using left and right (instead of just right) scattering data on different parts of the jump contour. Consequently it will be sufficient to approximate the left and right reflection coefficients. Details can be found in Section 6. Moreover, we obtain precise relations between the error terms and the decay of the initial conditions improving the error estimates obtained in Schuur [31] (which are stated in terms of smoothness and decay properties of R(k) and its derivatives). Overall we closely follow the recent review article [25], where Krüger and Teschl applied these methods to compute the long-time asymptotics for the Toda lattice. For a general result which applies in the case where R(k) has modulus strictly less than one and no solitons are present we refer to Varzugin [35] and for another recent generalization of the nonlinear steepest descent method to McLaughlin and Miller [28]. An alternate approach based on the asymptotic theory of pseudodifferential operators was given by Budylin and Buslaev [5]. Finally, note that if q(x, t) solves the KdV equation, then so does q(−x, −t). Therefore it suffices to investigate the case t → ∞.

2 The Inverse Scattering Transform and the Riemann–Hilbert Problem In this section we want to derive the Riemann–Hilbert problem for the KdV equation from scattering theory. This is essentially classical (compare, e.g., [4]) except for two points. The eigenvalues will be added by appropriate pole conditions which are then turned into jumps following Deift et al. [14]. We

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will impose an additional symmetry conditions to ensure uniqueness later on following Krüger and Teschl [24]. For the necessary results from scattering theory respectively the inverse scattering transform for the KdV equation we refer to [27] (see also [4] and [9]). We consider real-valued classical solutions q(x, t) of the KdV equation (1.1), which decay rapidly, that is  max (1 + |x|)|q(x, t)|dx < ∞, for all T > 0. (2.1) |t|T

R

Existence of such solutions can for example be established via the inverse scattering transform if one assumes (cf. [27, Sect. 4.2]) that the initial condition satisfies  (1 + |x|) (|q(x, 0)| + |qx (x, 0)| + |qxx (x, 0)| + |qxxx (x, 0)|) dx < ∞. (2.2) R

Associated with q(x, t) is a self-adjoint Schrödinger operator H(t) = −

d2 + q(., t), dx2

D(H) = H 2 (R) ⊂ L2 (R).

(2.3)

Here L2 (R) denotes the Hilbert space of square integrable (complex-valued) functions over R and H k (R) the corresponding Sobolev spaces. By our assumption (2.1) the spectrum of H consists of an absolutely continuous part [0, ∞) plus a finite number of eigenvalues −κj2 ∈ (−∞, 0), 1  j  N. In addition, there exist two Jost solutions ψ± (k, x, t) which solve the differential equation H(t)ψ± (k, x, t) = k2 ψ± (k, x, t),

Im(k) > 0,

(2.4)

and asymptotically look like the free solutions lim e∓ikx ψ± (k, x, t) = 1.

(2.5)

x→±∞

Both ψ± (k, x, t) are analytic for Im(k) > 0 and continuous for Im(k)  0. The asymptotics of the two Jost solutions are ψ± (k, x, t) = e

±ikx

 1 1 +O 1 + Q± (x, t) , 2ik k2



(2.6)

as k → ∞ with Im(k) > 0, where  Q+ (x, t) = −



∞

q(y, t)dy, x

Q− (x, t) = −

x

−∞

q(y, t)dy.

(2.7)

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Furthermore, one has the scattering relations T(k)ψ∓ (k, x, t) = ψ± (k, x, t) + R± (k, t)ψ± (k, x, t),

k ∈ R,

(2.8)

where T(k), R± (k, t) are the transmission respectively reflection coefficients. They have the following well-known properties: Lemma 2.1 The transmission coefficient T(k) is meromorphic for Im(k) > 0 with simple poles at iκ1 , . . . , iκ N and is continuous up to the real line. The residues of T(k) are given by 2 Resiκj T(k) = iμj(t)γ+, j(t)2 = iμjγ+, j,

(2.9)

where γ+, j(t)−1 = ψ+ (iκj, ., t)2

(2.10)

and ψ+ (iκj, x, t) = μj(t)ψ− (iκj, x, t). Moreover, T(k)R+ (k, t) + T(k)R− (k, t) = 0,

|T(k)|2 + |R± (k, t)|2 = 1.

(2.11)

In particular one reflection coefficient, say R(k, t) = R+ (k, t), and one set of norming constants, say γj(t) = γ+, j(t), suffices. Moreover, the time dependence is given by: Lemma 2.2 The time evolutions of the quantities R(k, t) and γ j(t) are given by 3

R(k, t) = R(k)e8ik t ,

(2.12)

3

γj(t) = γj e4κj t ,

(2.13)

where R(k) = R(k, 0) and γj = γj(0). We will set up a Riemann–Hilbert problem as follows:  m(k, x, t) =



T(k)ψ− (k, x, t)eikx

ψ+ (−k, x, t)eikx

ψ+ (k, x, t)e−ikx ,

T(−k)ψ− (−k, x, t)e

−ikx

Im(k) > 0, ,

Im(k) < 0. (2.14)

We are interested in the jump condition of m(k, x, t) on the real axis R (oriented from negative to positive). To formulate our jump condition we use the following convention: When representing functions on R, the lower subscript denotes the non-tangential limit from different sides. By m+ (k) we denote the limit from above and by m− (k) the one from below. Using the notation above implicitly assumes that these limits exist in the sense that m(k) extends to a continuous function on the real axis. In general, for an oriented

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contour , m+ (k) (resp. m− (k)) will denote the limit of m(κ) as κ → k from the positive (resp. negative) side of . Here the positive (resp. negative) side is the one which lies to the left (resp. right) as one traverses the contour in the direction of the orientation. Theorem 2.3 Let S+ (H(0)) = {R(k), k  0; (κj, γj), 1  j  N} be the right scattering data of the operator H(0). Then m(k) = m(k, x, t) defined in (2.14) is a solution of the following vector Riemann–Hilbert problem. Find a function m(k) which is meromorphic away from the real axis with simple poles at ±iκj and satisfies: (i) The jump condition



m+ (k) = m− (k)v(k),

v(k) =

for k ∈ R, (ii) the pole conditions

1 − |R(k)|2 R(k)et(k)

−R(k)e−t(k) , 1

0 0 Resiκj m(k) = lim m(k) , iγj2 et(iκj ) 0 k→iκj  0 −iγj2 et(iκj ) Res−iκj m(k) = lim m(k) , 0 0 k→−iκj

(2.15)



(iii) the symmetry condition

 m(−k) = m(k)

0 1 , 1 0

(2.16)

(2.17)

(iv) and the normalization lim m(iκ) = (1

κ→∞

1).

(2.18)

Here the phase is given by x (k) = 8ik3 + 2ik . t

(2.19)

Proof The jump condition (2.15) is a simple calculation using the scattering relations (2.8) plus (2.11). The pole conditions follow since T(k) is meromorphic for Im(k) > 0 with simple poles at iκj and residues given by (2.9). The symmetry condition holds by construction and the normalization (2.18) is immediate from the following lemma below.

Observe that the pole condition at iκj is sufficient since the one at −iκj follows by symmetry. Moreover, using  q(x, t) 1 T(k)ψ− (k, x, t)ψ+ (k, x, t) = 1 + + O (2.20) 2k2 k4

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as k → ∞ with Im(k) > 0 (observe that the right-hand side is just the diagonal Green’s functions of H(t) divided by the free one) we obtain from (2.6) Lemma 2.4 The function m(k, x, t) defined in (2.14) satisfies 

1 1 −1 1 + O m(k, x, t) = 1 1 + Q(x, t) . 2ik k2

(2.21)

Here Q(x, t) = Q+ (x, t) is defined in (2.7). For our further analysis it will be convenient to rewrite the pole condition as a jump condition and hence turn our meromorphic Riemann–Hilbert problem into a holomorphic Riemann–Hilbert problem following [14]. Choose ε so small that the discs |k − iκj| < ε lie inside the upper half plane and do not intersect. Then redefine m(k) in a neighborhood of iκj respectively −iκj according to ⎛ ⎞ ⎧ ⎪ 1 0 ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ m(k) ⎝ iγj2 et(iκj ) ⎠ , |k − iκj| < ε, ⎪ ⎪ ⎪ − 1 ⎪ ⎪ k − iκ j ⎪ ⎨ ⎛ ⎞ (2.22) m(k) = iγj2 et(iκj ) ⎪ ⎪ 1 ⎜ ⎟ ⎪ ⎪ m(k) ⎝ |k + iκj| < ε, k + iκj ⎠ , ⎪ ⎪ ⎪ ⎪ ⎪ 0 1 ⎪ ⎪ ⎩ m(k), else. Note that for Im(k) < 0 we redefined m(k) such that it respects our symmetry (2.17). Then a straightforward calculation using Resiκ m(k) = limk→iκ (k − iκ)m(k) shows: Lemma 2.5 Suppose m(k) is redefined as in (2.22). Then m(k) is holomorphic away from the real axis and the small circles around iκj and −iκj. Furthermore it satisfies (2.15), (2.17), (2.18) and the pole condition is replaced by the jump condition ⎛ ⎞ 1 0 2 t(iκj ) ⎠, |k − iκj| = ε, m+ (k) = m− (k) ⎝ iγj e − 1 k − iκj ⎛ m+ (k) = m− (k) ⎝1 0

−

iγj2 et(iκj )

⎞

k + iκj ⎠ , 1

|k + iκj| = ε,

(2.23)

where the small circle around iκj is oriented counterclockwise and the one around −iκj is oriented clockwise. Next we turn to uniqueness of the solution of this vector Riemann–Hilbert problem. This will also explain the reason for our symmetry condition. We

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begin by observing that if there is a point k1 ∈ C, such that m(k1 ) = 0 0 , 1 m(k) satisfies the same jump and pole conditions as m(k). then n(k) = k−k 1 However, it will clearly violate the symmetry condition! Hence, without the symmetry condition, the solution of our vector Riemann–Hilbert problem will not be unique in such a situation. Moreover, a look at the one-soliton solution verifies that this case indeed can happen. Lemma 2.6 (One-soliton solution) Suppose there is only one eigenvalue and that the reflection coefficient vanishes, that is, S+ (H(t)) = {R(k, t) ≡ 0, k ∈ R; (κ, γ (t)), κ > 0, γ > 0}. Then the unique solution of the Riemann–Hilbert problem (2.15)–(2.18) is given by

m0 (k) = f (k) f (−k) f (k) =



1 1 + (2κ)−1 γ 2 et(iκ)

1+

k + iκ (2κ)−1 γ 2 et(iκ) . k − iκ

(2.24)

Furthermore, the zero solution is the only solution of the corresponding vanishing problem where the normalization is replaced by limk→∞ m(ik) = (0 0). In particular, Q(x, t) =

2γ 2 et(iκ) . 1 + (2κ)−1 γ 2 et(iκ)

(2.25)

Proof By assumption the reflection coefficient vanishes and so the jump along the real axis disappears. Therefore and by the symmetry

condition, we know that the solution is of the form m0 (k) = f (k) f (−k) where f (k) is meromorphic. Furthermore the function f (k) has only a simple pole at iκ, so that we can make the ansatz f (k) = C + D k+iκ . Then the constants C and D k−iκ are uniquely determined by the pole conditions and the normalization.

2 t(iκ) In fact, observe f (k1 ) = f (−k1 ) = 0 if and only . if k 1 = 0 and 2κ = γ e Furthermore, even in the general case m(k1 ) = 0 0 can only occur at k1 = 0 as the following lemma shows.



Lemma 2.7 If m(k1 ) = 0 0 for m defined as in (2.14), then k1 = 0. Moreover, the zero of at least one component is simple in this case.

Proof By (2.14) the condition m(k1 ) = 0 0 implies that the Jost solutions ψ− (k, x) and ψ+ (k, x) are linearly dependent or that the transmission coefficient T(k1 ) = 0. This can only happen, at the band edge, k1 = 0 or at an eigenvalue k1 = iκj. We begin with the case k1 = iκj. In this case the derivative of the Wronskian W(k) = ψ+ (k, x)ψ− (k, x) − ψ+ (k, x)ψ− (k, x) does not vanish by the well d known formula dk W(k)|k=k1 = −2k1 R ψ+ (k1 , x)ψ− (k1 , x)dx = 0. Moreover,

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the diagonal Green’s function g(z, x) = W(k)−1 ψ+ (k, x)ψ− (k, x) is Herglotz as a function of z = −k2 and hence can have at most a simple zero at z = −k21 . Since z → −k2 is conformal away from z = 0 the same is true as a function of k. Hence, if ψ+ (iκj, x) = ψ− (iκj, x) = 0, both can have at most a simple zero at k = iκj. But T(k) has a simple pole at iκj and hence T(k)ψ− (k, x) cannot vanish at k = iκj, a contradiction. It remains to show that one zero is simple in the case k1 = 0. In fact, one d can show that dk W(k)|k=k1 = 0 in this case as follows: First of all note that ψ˙ ± (k) (where the dot denotes the derivative with respect to k) again solves H ψ˙ ± (k1 ) = −k21 ψ˙ ± (k1 ) if k1 = 0. Moreover, by W(k1 ) = 0 we have ψ+ (k1 ) = c ψ− (k1 ) for some constant c (independent of x). Thus we can compute ˙ 1 ) = W(ψ˙ + (k1 ), ψ− (k1 )) + W(ψ+ (k1 ), ψ˙ − (k1 )) W(k = c−1 W(ψ˙ + (k1 ), ψ+ (k1 )) + cW(ψ− (k1 ), ψ˙ − (k1 )) by letting x → +∞ for the first and x → −∞ for the second Wronskian (in which case we can replace ψ± (k) by e±ikx ), which gives ˙ 1 ) = −i(c + c−1 ). W(k Hence the Wronskian has a simple zero. But if both functions had more than simple zeros, so would the Wronskian, a contradiction.



3 A Uniqueness Result for Symmetric Vector Riemann–Hilbert Problems In this section we want to investigate uniqueness for the holomorphic vector Riemann–Hilbert problem m+ (k) = m− (k)v(k),  0 1 m(−k) = m(k) , 1 0

lim m(iκ) = 1 1 ,

κ→∞

k ∈ ,

(3.1)

where we assume Hypothesis 3.1 Let consist of a finite number of smooth oriented curves in C such that the distance between and {iy|y  y0 } is positive for some y0 > 0. Assume that the contour is invariant under k → −k and v(k) is symmetric   0 1 0 1 , k ∈ . (3.2) v(−k) = v(k)−1 1 0 1 0 Moreover, suppose det(v(k)) = 1.

Long-Time Asymptotics for the KdV Equation

297

Now we are ready to show that the symmetry condition in fact guarantees uniqueness. Theorem 3.2 Assume Hypothesis 3.1. Suppose there exists a solution m(k) of the Riemann–Hilbert problem (3.1) for which m(k) = 0 0 can happen at most for k = 0 in which case lim supk→0 m jk(k) is bounded from any direction for j = 1 or j = 2. Then the Riemann–Hilbert problem (3.1) with norming condition replaced by

lim m(iκ) = α α (3.3) κ→∞

for given α ∈ C, has a unique solution mα (k) = α m(k). Proof Let mα (k) be a solution of (3.1) normalized according to (3.3). Then we can construct a matrix valued solution via M = (m, mα ) and there are two possible cases: Either det M(k) is nonzero for some k or it vanishes identically. We start with the first case. Since the determinant of our Riemann–Hilbert problem has no jump and is bounded at infinity, it is constant. But taking determinants in  0 1 M(−k) = M(k) . 1 0 gives a contradiction. It remains to investigate the case where det(M) ≡ 0. In this case we have mα (k) = δ(k)m(k) with a scalar function δ. Moreover, δ(k) must be holomorphic for k ∈ C\

and continuous for k ∈ except possibly at the points where m(k1 ) = 0 0 . Since it has no jump across , δ+ (k)m+ (k) = mα,+ (k) = mα,− (k)v(k) = δ− (k)m− (k)v(k) = δ− (k)m+ (k), it is even holomorphic in C\{0} with at most a simple pole at k = 0. Hence it must be of the form δ(k) = A +

B . k

Since δ has to be symmetric, δ(k) = δ(−k), we obtain B = 0. Now, by the normalization we obtain δ(k) = A = α. This finishes the proof.

Furthermore, the requirements cannot be relaxed to allow (e.g.) second order zeros in stead of simple zeros. In fact, if m(k) is a solution for which ˜ both components vanish of second order at, say, k = 0, then m(k) = k12 m(k) is a nontrivial symmetric solution of the vanishing problem (i.e. for α = 0). By Lemma 2.7 we have Corollary 3.3 The solution m(k) = m(k, x, t) found in Theorem 2.3 is the only solution of the vector Riemann–Hilbert problem (2.15–2.18).

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Observe that there is nothing special about k → ∞ where we normalize, any other point would do as well. However, observe that for the one-soliton solution (2.24), f (k) vanishes at k = iκ

1 − (2κ)2 γ 2 et(iκ) 1 + (2κ)2 γ 2 et(iκ)

and hence the Riemann–Hilbert problem normalized at this point has a nontrivial solution for α = 0 and hence, by our uniqueness result, no solution for α = 1. This shows that uniqueness and existence are connected, a fact which is not surprising since our Riemann–Hilbert problem is equivalent to a singular integral equation which is Fredholm of index zero (see Appendix A).

4 Conjugation and Deformation This section demonstrates how to conjugate our Riemann–Hilbert problem and how to deform our jump contour, such that the jumps will be exponentially close to the identity away from the stationary phase points. Throughout this and the following section, we will assume that the R(k) has an analytic extension to a small neighborhood of the real axis. This is for example the case if we assume that our solution is exponentially decaying. In Section 6 we will show how to remove this assumption. For easy reference we note the following result:  ⊆ . Let D be a matrix of the form Lemma 4.1 (Conjugation) Assume that

 0 d(k)−1 , (4.1) D(k) = 0 d(k)  → C is a sectionally analytic function. Set where d : C\

˜ m(k) = m(k)D(k),

(4.2)

then the jump matrix transforms according to v(k) ˜ = D− (k)−1 v(k)D+ (k).

(4.3)

If d satisfies d(−k) = d(k)−1 and limκ→∞ d(iκ) = 1, then the transformation ˜ ˜ m(k) = m(k)D(k) respects our symmetry, that is, m(k) satisfies (2.17) if and only if m(k) does, and our normalization condition. In particular, we obtain  v11 v˜ = v21 d−2

v12 d2 , v22

, k ∈ \

(4.4)

Long-Time Asymptotics for the KdV Equation

299

respectively ⎞

⎛

d− ⎜ d v11 + v˜ = ⎜ ⎝ −1 v21 d−1 + d−

v12 d+ d− ⎟ ⎟, ⎠ d+ v22 d−

. k∈

(4.5)

In order to remove the poles there are two cases to distinguish. If Re((iκ j)) < 0, then the corresponding jump is exponentially close to the identity as t → ∞ and there is nothing to do. Otherwise we use conjugation to turn the jumps into one with exponentially decaying off-diagonal entries, again following Deift et al. [14]. It turns out that we will have to handle the poles at iκj and −iκj in one step in order to preserve symmetry and in order to not add additional poles elsewhere. Lemma 4.2 Assume that the Riemann–Hilbert problem for m has jump conditions near iκ and −iκ given by ⎛

⎞ 1 0 ⎠, m+ (k) = m− (k) ⎝ iγ 2 − 1 k − iκ ⎞ ⎛ iγ 2 m+ (k) = m− (k) ⎝1 − k + iκ ⎠ , 0 1

|k − iκ| = ε,

|k + iκ| = ε.

(4.6)

Then this Riemann–Hilbert problem is equivalent to a Riemann–Hilbert prob˜ which has jump conditions near iκ and −iκ given by lem for m ⎛

⎞ (k + iκ)2 ˜ + (k) = m ˜ − (k) m iγ 2 (k − iκ) ⎠ , 0 1 ⎛ ⎞ 1 0 ⎠, ˜ − (k) ⎝ (k − iκ)2 ˜ + (k) = m m − 2 1 iγ (k + iκ) ⎝1

−

|k − iκ| = ε,

|k + iκ| = ε,

and all remaining data conjugated (as in Lemma 4.1) by ⎛

k − iκ ⎜ k + iκ D(k) = ⎝ 0

⎞ 0

⎟ k + iκ ⎠ . k − iκ

(4.7)

300

K. Grunert, G. Teschl

Proof To turn γ 2 into γ −2 , introduce D by ⎞ ⎧⎛ k − iκ ⎛ k − iκ ⎪ ⎪ − ⎪ ⎜ 1 ⎪ ⎪ iγ 2 ⎟ k + iκ ⎜ ⎟⎜ ⎪ ⎪ 2 ⎝ ⎠⎝ ⎪ iγ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ k − iκ ⎪ ⎪ ⎪ ⎞⎛ ⎛ ⎪ ⎪ 2 ⎪ iγ k − iκ ⎪ ⎨⎜ 0 − ⎟ k + iκ ⎟ ⎜ k + iκ D(k) = ⎜ ⎠⎝ ⎝ k + iκ ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ iγ 2 ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ k − iκ ⎪ ⎪ ⎪ 0 ⎪ ⎜ k + iκ ⎟ ⎪ ⎪ ⎪ ⎪⎝ k + iκ ⎠ , ⎪ ⎩ 0 k − iκ

⎞ 0

⎟ k + iκ ⎠ , k − iκ ⎞ 0 ⎟ k + iκ ⎠ ,

|k − iκ| < ε,

|k + iκ| < ε,

k − iκ

else,

˜ and note that D(k) is analytic away from the two circles. Now set m(k) = m(k)D(k)and note that D(k) is also symmetric. Therefore the jump conditions can be verified by straightforward calculations and Lemma 4.1.

The jump along the real axis is of oscillatory type and our aim is to apply a contour deformation following [10] such that all jumps will be moved into regions where the oscillatory terms will decay exponentially. Since the jump matrix v contains both exp(t) and exp(−t) we need to separate them in order to be able to move them to different regions of the complex plane. We recall that the phase of the associated Riemann–Hilbert problem is given by (k) = 8ik3 + 2ik

x t

(4.8)

and the stationary phase points,  (k) = 0, are denoted by ±k0 , where  x . k0 = − 12t For xt > 0 we have k0 ∈ iR, and for xt < 0 we have k0 ∈ R. For also need the value κ0 defined via Re((iκ0 )) = 0, that is,  x κ0 = > 0. 4t

x t

(4.9)

> 0 we will

(4.10)

We will set κ0 = 0 if xt < 0 for notational convenience. A simple analysis shows that for xt > 0 we have 0 < k0 /i < κ0 . As mentioned above we will need the following factorizations of the jump condition (2.15): v(k) = b − (k)−1 b + (k),

(4.11)

Long-Time Asymptotics for the KdV Equation

where

  1 R(k)e−t(k) b − (k) = , 0 1

for |k| > Re(k0 ) and

⎛

v(k) = B− (k)−1 ⎝ where

⎛

B− (k) = ⎝ −

1 R(k)et(k) 1 − |R(k)|

2

0 1

301

 b + (k) =

1 − |R(k)|2 0

⎞

1 0 . R(k)et(k) 1 ⎞

0 1

⎠ B+ (k),

1 − |R(k)|

(4.13)

2

⎞ ⎛ R(k)e−t(k) 1 − ⎠ B+ (k) = ⎝ 1 − |R(k)|2 . 0 1

⎠,

(4.12)

(4.14)

for |k| < Re(k0 ). To get rid of the diagonal part in the factorization corresponding to |k| < Re(k0 ) and to conjugate the jumps near the eigenvalues we need the partial transmission coefficient T(k, k0 ). We define the partial transmission coefficient with respect to k0 by ⎧  k + iκ j ⎪ ⎪ , k0 ∈ iR+ , ⎪ ⎪ ⎪ k − iκ j ⎪ ⎨κ j ∈(κ0 ,∞) ⎛ ⎞ (4.15) T(k, k0 ) = N k0 2  ⎪ k + iκ j log(|T(ζ )| ) ⎟ ⎜ 1 ⎪ + ⎪ exp dζ , k ∈ R , ⎪ ⎝ ⎠ 0 ⎪ ⎪ 2π i ζ −k ⎩ j=1 k − iκ j −k0

for k ∈ C\ (k0 ), where (k0 ) = [− Re(k0 ), Re(k0 )] (oriented from left to right). Thus T(k, k0 ) is meromorphic for k ∈ C\ (k0 ). Note that T(k, k0 ) can be computed in terms of the scattering data since |T(k)|2 = 1 − |R+ (k, t)|2 . Moreover, we set ⎧  ⎪ 2κ j, k0 ∈ iR+ , ⎪ ⎪ ⎪ ⎨κ j ∈(κ0 ,∞)  k0 N (4.16) T1 (k0 ) =  1 ⎪ 2 + ⎪ 2κ + log(|T(ζ )| )dζ, k ∈ R . ⎪ j 0 ⎪ ⎩ 2π j=1

Thus

−k0

 1 i T(k, k0 ) = 1 + T1 (k0 ) + O , k k2

as k → ∞.

(4.17)

Theorem 4.3 The partial transmission coefficient T(k, k0 ) is meromorphic in C\ (k0 ), where

(k0 ) = [− Re(k0 ), Re(k0 )],

(4.18)

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K. Grunert, G. Teschl

with simple poles at iκ j and simple zeros at −iκ j for all j with κ0 < κ j, and satisfies the jump condition T+ (k, k0 ) = T− (k, k0 )(1 − |R(k)|2 ),

for k ∈ (k0 ).

(4.19)

Moreover, (i) T(−k, k0 ) = T(k, k0 )−1 , k ∈ C\ (k0 ), (ii) T(−k, k0 ) = T(k, k0 ), k ∈ C, in particular T(k, k0 ) is real for k ∈ iR, and (iii) if k0 ∈ R+ the behaviour near k = 0 is given by T(k, k0 ) = T(k)(C + o(1)) with C = 0 for Im(k)  0. Proof That iκ j are simple poles and −iκ j are simple zeros is obvious from the Blaschke factors and that T(k, k0 ) has the given jump follows from Plemelj’s formulas. (i), (ii), and (iii) are straightforward to check.

Now we are ready to perform our conjugation step. Introduce ⎞ ⎧⎛ k − iκ j ⎪ ⎪ 1 − 2 t(iκ ) ⎟ ⎪ ⎜ ⎪ j ⎪ iγ j e ⎜ ⎟ ⎪ ⎪ ⎜ ⎟ D0 (k), ⎪ 2 t(iκ ) j ⎪ iγ e ⎝ ⎠ ⎪ j ⎪ ⎪ 0 ⎪ ⎪ k − iκ j ⎪ ⎪ ⎪ ⎨⎛ ⎞ iγ j2 et(iκ j ) D(k) = ⎪ 0 − ⎟ ⎜ ⎪ ⎪ ⎜ k + iκ j ⎟ ⎪ ⎪ ⎟ D0 (k), ⎜ ⎪ ⎪ ⎠ ⎝ k + iκ j ⎪ ⎪ 1 ⎪ 2 t(iκ j ) ⎪ ⎪ iγ e ⎪ j ⎪ ⎪ ⎪ ⎪ ⎩ D0 (k),

|k − iκ j| < ε,

κ0 < κ j ,

|k + iκ j| < ε,

κ0 < κ j ,

else, (4.20)

where 

T(k, k0 )−1 0 D0 (k) = . 0 T(k, k0 ) Observe that D(k) respects our symmetry,   0 1 0 1 D(−k) = D(k) . 1 0 1 0 Now we conjugate our problem using D(k) and set ˜ m(k) = m(k)D(k).

(4.21)

Note that even though D(k) might be singular at k = 0 (if k0 > 0 and ˜ R(0) = −1), m(k) is nonsingular since the possible singular behaviour of T(k, k0 )−1 from D0 (k) cancels with T(k) in m(k) by virtue of Theorem 4.3 (iii).

Long-Time Asymptotics for the KdV Equation

303

Then using Lemmas 4.1 and 4.2 the jump corresponding to κ0 < κ j (if any) is given by ⎛ ⎞ k − iκ j 1 − 2 t(iκ ) j T(k, k )−2 ⎠ , v(k) ˜ =⎝ |k − iκ j| = ε, iγ j e 0 0 1 ⎛ ⎞ 1 0 k + iκ j ⎠ v(k) ˜ = ⎝− |k + iκ j| = ε, (4.22) 1 , 2 t(iκ j ) iγ j e T(k, k0 )2 and corresponding to κ0 > κ j (if any) by ⎛ ⎞ 1 0 2 t(iκ ) −2 j ⎠, T(k, k0 ) v(k) ˜ = ⎝ iγ j e − 1 k − iκ j ⎛ ⎞ iγ j2 et(iκ j ) T(k, k0 )2 ⎠, v(k) ˜ = ⎝1 − k + iκ j 0 1

|k − iκ j| = ε,

|k + iκ j| = ε.

(4.23)

In particular, all jumps corresponding to poles, except for possibly one if κ j = κ0 , are exponentially close to the identity for t → ∞. In the latter case we will keep the pole condition for κ j = κ0 which now reads  0 0 ˜ ˜ = limk→iκ j m(k) , Resiκ j m(k) iγ j2 et(iκ j ) T(iκ j, k0 )−2 0  0 −iγ j2 et(iκ j ) T(iκ j, k0 )−2 ˜ ˜ = limk→−iκ j m(k) . (4.24) Res−iκ j m(k) 0 0 Furthermore, the jump along R is given by  b˜ − (k)−1 b˜ + (k), v(k) ˜ = B˜ − (k)−1 B˜ + (k), where

⎛

1 b˜ − (k) = ⎝ 0 and

⎞ R(−k)e−t(k) T(−k, k0 )2 ⎠ , 1

k ∈ R\ (k0 ), k ∈ (k0 ), ⎛

1 b˜ + (k) = ⎝ R(k)et(k) T(k, k0 )2

(4.25)

0 1

⎞ ⎠,

(4.26)

⎞ ⎛ ⎞ 1 0 1 0 ⎠ = ⎝ T− (−k, k0 ) ⎠, B˜ − (k) = ⎝ T− (k, k0 )−2 − R(k)et(k) 1 R(k)et(k) 1 − 2 T− (k, k0 ) 1 − |R(k)| ⎞ ⎛ ⎞ ⎛ T+ (k, k0 ) T+ (k, k0 )2 −t(k) −t(k) R(−k)e 1 − R(−k)e 1 − ⎠. ⎠=⎝ B˜ + (k) = ⎝ T+ (−k, k0 ) 1 − |R(k)|2 0 1 0 1 ⎛

304

K. Grunert, G. Teschl

Here we have used R(−k) = R(k),

k ∈ R,

T± (−k, k0 ) = T∓ (k, k0 )−1 ,

k ∈ (k0 ),

and the jump condition (4.19) for the partial transmission coefficient T(k, k0 ) along (k0 ) in the last step. This also shows that the matrix entries are bounded for k ∈ R near k = 0 since T± (−k, k0 ) = T± (k, k0 ). Since we have assumed that R(k) has an analytic continuation to a neighborhood of the real axis, we can now deform the jump along R to move the oscillatory terms into regions where they are decaying. According to Fig. 2 there are two cases to distinguish: Case 1 k0 ∈ iR, k0 = 0 We set ± = {k ∈ C| Im(k) = ±ε} for some small ε such that ± lies in the region with ± Re((k)) < 0 and such that the circles around ±iκ j lie outside the region in between − and + (see Fig. 3). Then we can split our jump by ˜ redefining m(k) according to ⎧ ⎪ ˜ b˜ + (k)−1 , 0 < Im(k) < ε, ⎨m(k) (k) = m(k) m (4.27) ˜ b˜ − (k)−1 , −ε < Im(k) < 0, ⎪ ⎩ ˜ m(k), else. Thus the jump along the real axis disappears and the jump along ± is given by  k ∈ + b˜ + (k), (4.28)  v (k) = k ∈ − . b˜ − (k)−1 , All other jumps are unchanged. Note that the resulting Riemann–Hilbert problem still satisfies our symmetry condition (2.17), since we have   0 1 ˜ 0 1 ˜ b ± (−k) = b ∓ (k) . 1 0 1 0

Fig. 2 Sign of Re((k)) for different values of k0

(4.29)

Long-Time Asymptotics for the KdV Equation

305

Fig. 3 Deformed contour for k0 ∈ iR+

By construction the jump along ± is exponentially close to the identity as t → ∞. Case 2 k0 ∈ R, k0 = 0 We set ± = ±1 ∪ ±2 according to Fig. 4 chosen such that the circles around ±iκ j lie outside the region in between − and + . Again note that

±1 respectively ±2 lie in the region with ± Re((k)) < 0. Then we can split ˜ our jump by redefining m(k) according to ⎧ ⎪ ˜ m(k) b˜ + (k)−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ˜ b˜ − (k)−1 , ⎨m(k) (k) = m(k) m ˜ B˜ + (k)−1 , ⎪ ⎪ ⎪ ˜ m(k) B˜ − (k)−1 , ⎪ ⎪ ⎪ ⎩m(k), ˜

Fig. 4 Deformed contour for k0 ∈ R+

k between R and +1 , k between R and −1 , k between R and +2 , k between R and −2 , else.

(4.30)

306

K. Grunert, G. Teschl

One checks that the jump along R disappears and the jump along ± is given by ⎧ b˜ + (k), ⎪ ⎪ ⎪ ⎨b˜ (k)−1 , −  v (k) = ˜ ⎪ B + (k), ⎪ ⎪ ⎩˜ B− (k)−1 ,

k ∈ +1 , k ∈ −1 , k ∈ +2 , k ∈ −2 .

(4.31)

All other jumps are unchanged. Again the resulting Riemann–Hilbert problem still satisfies our symmetry condition (2.17) and the jump along

± \{k0 , −k0 } is exponentially decreasing as t → ∞ Theorem 4.4 Assume

 R

(1 + |x|)1+l |q(x, 0)|dx < ∞

(4.32)

for some integer l  1 and abbreviate by c j = 4κ 2j the velocity of the j’th soliton determined by Re((iκ j)) = 0. Then the asymptotics in the soliton region, x/t  C for some C > 0, are as follows: Let ε > 0 sufficiently small such that the intervals [c j − ε, c j + ε], 1  j  N, are disjoint and lie inside R+ . If | xt − c j| < ε for some j, one has 

∞

N 

q(y, t)dy = −4

x

κi −

i= j+1

2γ j2 (x, t) 1 + (2κ j)−1 γ j2 (x, t)

+ O(t−l ),

(4.33)

respectively q(x, t) =

−4κ jγ j2 (x, t)

+ O(t−l ),

(4.34)

N   κi − κ j 2 . κi + κ j i= j+1

(4.35)

(1 + (2κ j)−1 γ j2 (x, t))2

where γ j2 (x, t) = γ j2 e−2κ j x+8κ j t 3

If | xt − c j|  ε, for all j, one has 

∞ x

q(y, t)dy = −4



 −l

κi + O(t ),

κi ∈(κ0 ,∞)

κ0 =

x , 4t

(4.36)

respectively q(x, t) = O(t−l ).

(4.37)

Long-Time Asymptotics for the KdV Equation

307

˜ (k) = m(k) Proof Since m for k sufficiently far away from R equations (2.21), (4.21), and (4.17) imply the following asymptotics 

1 1 −1 1 + O (k) = 1 1 + (−2T1 (k0 ) + Q(x, t)) m . (4.38) 2ik k2 By construction, the jump along ± is exponentially decreasing as t → ∞. Hence we can apply Theorem A.6 as follows:  If | xt − c j| > ε (resp. |κ02 − κ 2j | > ε) for all j we can choose γ t = 0 and wt = w in Theorem A.6. Since w  is exponentially small as t → ∞, the solutions of the associated Riemann–Hilbert problems only differ by O(t−l ) for any

l  1. Comparing m0 = 1 1 with the above asymptotics shows Q+ (x, t) = 2T1 (k0 ) + O(t−l ). If | xt − c j| < ε (resp. |κ02 − κ 2j | < ε) for some j, we choose γ t = γk (x, t) and wt = w  in Theorem A.6, where  N   κ j −2 κi − κ j 2 3 = γ j2 e−2κ j x+8κ j t . γ j2 (x, t) = γ j2 et(iκ j ) T iκ j, i √ κi + κ j 3 i= j+1 As before we conclude that w  is exponentially small and so the associated solutions of the Riemann–Hilbert problems only differ by O(t −l ). From Lemma 2.6, we have the one-soliton solution m0 (k) = f (k) f (−k) with  k + iκ j 1 −1 2 f (k) = (2κ j) γ j (x, t) . 1+ k − iκ j 1 + (2κ j)−1 γ j2 (x, t) As before, comparing with the above asymptotics shows Q(x, t) = 2T1 (k0 ) +

2γ j2 (x, t) 1 + (2κ j)−1 γ j2 (x, t)

+ O(t−l ).

To see the second part just use (2.20) in place of (2.21). This finishes the proof in the case where R(k) has an analytic extensions. We will remove this assumption in Section 6 thereby completing the proof.

Since the one-soliton solution is exponentially decaying away from its minimum, we also obtain the form stated in the introduction: Corollary 4.5 Assume (4.32), then the asymptotic in the soliton region, x/t  C for some C > 0, is given by q(x, t) = −2 where

N 

κ 2j

j=1

cosh2 (κ j x − 4κ 3j t − p j)

+ O(t−l ),

⎞ 2 N  2  γ − κ κ 1 i j j ⎠. p j = log ⎝ 2 2κ j i= j+1 κi + κ j

(4.39)

⎛

(4.40)

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K. Grunert, G. Teschl

5 Reduction to a Riemann–Hilbert Problem on a Small Cross In the previous section we have seen that for k0 ∈ R we can reduce everything (k) such that the jumps are exponentially to a Riemann–Hilbert problem for m close to the identity except in small neighborhoods of the stationary phase points k0 and −k0 . Hence we need to continue our investigation of this case in this section. Denote by c (±k0 ) the parts of + ∪ − inside a small neighborhood of ±k0 . We will now show that solving the two problems on the small crosses c (k0 ) respectively c (−k0 ) will lead us to the solution of our original problem. In fact, the solution of both crosses can be reduced to the following model problem: Introduce the cross = 1 ∪ · · · ∪ 4 (see Fig. 5) by

1 = {ue−iπ/4 , u ∈ [0, ∞)}

2 = {ueiπ/4 , u ∈ [0, ∞)}

3 = {ue3iπ/4 , u ∈ [0, ∞)}

4 = {ue−3iπ/4 , u ∈ [0, ∞)}.

(5.1)

Orient such that the real part of k increases in the positive direction. Denote by D = {ζ, |ζ | < 1} the open unit disc. Throughout this section ζ iν will denote the function eiν log(ζ ) , where the branch cut of the logarithm is chosen along the negative real axis (−∞, 0). Introduce the following jump matrices (v j for ζ ∈ j)   1 0 1 −R1 (ζ )ζ 2iν e−t(ζ ) v1 = , v2 = , 0 1 R2 (ζ )ζ −2iν et(ζ ) 1   1 0 1 −R3 (ζ )ζ 2iν e−t(ζ ) , v4 = (5.2) v3 = R4 (ζ )ζ −2iν et(ζ ) 1 0 1

Fig. 5 Contours of a cross

Long-Time Asymptotics for the KdV Equation

309

and consider the RHP given by M+ (ζ ) = M− (ζ )v j(ζ ), M(ζ ) → I,

ζ ∈ j, ζ → ∞.

j = 1, 2, 3, 4, (5.3)

The solution is given in the following theorem of Deift and Zhou [10] (for a proof of the version stated below see Krüger and Teschl [25]). Theorem 5.1 ([10]) Assume there is some ρ0 > 0 such that v j(ζ ) = I for |ζ | > ρ0 . Moreover, suppose that within |ζ |  ρ0 the following estimates hold: (i) The phase satisfies (0) = i0 ∈ iR,  (0) = 0,  (0) = i and 

1 2 + for ζ ∈ 1 ∪ 3 , ± Re (ζ )  |ζ | , 4 − else, |(ζ ) − (0) −

iζ 2 |  C|ζ |3 . 2

(5.4)

(5.5)

(ii) There is some r ∈ D and constants (α, L) ∈ (0, 1] × (0, ∞) such that R j, j = 1, . . . , 4, satisfy Hölder conditions of the form |R1 (ζ ) − r|  L|ζ |α , |R3 (ζ ) −

r |  L|ζ |α , 1 − |r|2

|R2 (ζ ) − r|  L|ζ |α , |R4 (ζ ) −

r |  L|ζ |α . 1 − |r|2

Then the solution of the RHP (5.3) satisfies  1+α 1 i 0 −β M(ζ ) = I + 1/2 + O t− 2 , β 0 ζt

(5.6)

(5.7)

for |ζ | > ρ0 , where β=

√ i(π/4−arg(r)+arg((iν))) −it0 −iν e t , νe

ν=−

1 log(1 − |r|2 ). 2π

(5.8)

Furthermore, if R j(ζ ) and (ζ ) depend on some parameter, the error term is uniform with respect to this parameter as long as r remains within a compact subset of D and the constants in the above estimates can be chosen independent of the parameters. Theorem 5.2 (Decoupling) Consider the Riemann–Hilbert problem m+ (k) = m− (k)v(k),

lim m(iκ) = 1 1 , κ→∞

k ∈ , (5.9)

with det(v) = 0 and let 0 < α < β  2α, ρ(t) → ∞ be given. Suppose that for every sufficiently small ε > 0 both the L2 and the L∞ norms of v are O(t−β ) away from some ε neighborhoods of some points k j ∈ ,

310

K. Grunert, G. Teschl

1  j  n. Moreover, suppose that the solution of the matrix problem with jump v(k) restricted to the ε neighborhood of k j has a solution which satisfies M j(k) = I +

Mj 1 + O(ρ(t)−β ), ρ(t)α k − k j

  k − k j > ε.

(5.10)

Then the solution m(k) is given by

m(k) = 1 1 +

n

 Mj 1 1 1 + O(ρ(t)−β ), α ρ(t) k − k j j=1

(5.11)

where the error term depends on the distance of k to . Proof In this proof we will use the theory developed in Appendix A with ds m0 (k) = 1 1 and the usual Cauchy kernel ∞ (s, k) = I s−k . Moreover, since 2 symmetry is not important, we will consider Cw on L ( ) rather than restricting it to the symmetric subspace L2s ( ). Here w± = ±(b ± − I) correspond to some factorization v = b −1 − b + of v (e.g., b − = I and b + = v). Assume that m(k) exists, then the same arguments as in the appendix show that

1 m(k) = 1 1 + 2π i



μ(s)(w+ (s) + w− (s))∞ (s, k),

where μ solves



(I − Cw )(μ − 1 1 ) = Cw 1 1 . ˆ Introduce m(k) by



m(k)M j(k)−1 , ˆ m(k) = m(k),

  k − k j  2ε, else.

ˆ The Riemann–Hilbert problem for m(k) has jumps given by   ⎧ −1 k − k j = 2ε, ⎪ ⎪ M j(k) ,   ⎪ ⎨ M (k)v(k)M (k)−1 , k ∈ , ε < k − k j < 2ε, j j   v(k) ˆ = ⎪ I, k ∈ , k − k j < ε, ⎪ ⎪ ⎩ v(k), else.

(5.12)

(5.13)

By assumption the jumps are I + O(ρ(t)−α ) on the circles |k − k j| = 2ε and even I + O(ρ(t)−β ) on the rest (both in the L2 and L∞ norms). In particular, we infer that (I − Cwˆ )−1 exists for

sufficiently large t and

using the Neumann series to estimate (μˆ − 1 1 ) = (I − Cwˆ )−1 Cwˆ 1 1 (cf. the proof of Theorem A.6) we obtain 

 ˆ 2 μˆ − 1 1   cw = O(ρ(t)−α ). (5.14) 2 1 − cw ˆ ∞

Long-Time Asymptotics for the KdV Equation

311

Thus we conclude 

1 ds m(k) = 1 1 + μ(s) ˆ w(s) ˆ 2π i ˆ s−k n 

1  ds −1 + O ρ(t)−β = 1 1 + μ(s)(M ˆ − I) j (s) 2π i j=1 |s−k j |=2ε s−k  n

1 



1 ds −α 1 1 Mj = 1 1 − ρ(t) + O ρ(t)−β 2π i j=1 s − k s − k j |s−k j |=2ε

−α

= 1 1 + ρ(t)



1 1

n

 j=1



Mj + O ρ(t)−β , k − kj (5.15)



and hence the claim is proven.

Now let us turn to the solution of the problem on c (k0 ) = ( + ∪ − ) ∩ {k| |k − k0 | < ε} for some small ε > 0. Without loss we can also deform our contour slightly such that c (k0 ) consists of two straight lines. Next, note  (k0 ) = 48ik0 .

(k0 ) = −16ik30 ,

As a first step we make a change of coordinates  ζ ζ = 48k0 (k − k0 ), k = k0 + √ 48k0

(5.16)

such that the phase reads (k) = (k0 ) + 2i ζ 2 + O(ζ 3 ). Next we need the behavior of our jump matrix near k0 , that is, the behavior of T(k, k0 ) near k0 . Lemma 5.3 Let k0 ∈ R, then T(k, k0 ) =



k − k0 k + k0

iν

˜ T(k, k0 ),

(5.17)

where ν = − π1 log(|T(k0 )|) > 0 and the branch cut of the logarithm is chosen along the negative real axis. Here ⎛ ⎞   k0 N 2  k + iκ j |T(ζ )| 1 ⎜ 1 ⎟ ˜ exp ⎝ log T(k, k0 ) = dζ ⎠ (5.18) 2 k − iκ j 2π i |T(k0 )| ζ − k j=1 −k0

is Hölder continuous of any exponent less than 1 at the stationary phase point ˜ 0 , k0 )| = 1. k = k0 and satisfies |T(k Proof First of all observe that     k0 1 1 k − k0 iν 2 exp log(|T(k0 )| ) . dζ = 2π i −k0 ζ −k k + k0 Hölder continuity of any exponent less than 1 is well-known (cf. [29]).

(5.19)



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K. Grunert, G. Teschl

If k(ζ ) is defined as in (5.16) and 0 < α < 1, then there is an L > 0 such that   √  ˜ 0 , k0 )e−iν log(2k0 48k0 )   L |ζ |α , T(k(ζ ), k0 ) − ζ iν T(k

(5.20)

where the branch cut of ζ iν is chosen along the negative real axis. We also have |R(k(ζ )) − R(k0 )|  L |ζ |α

(5.21)

and thus the assumptions of Theorem 5.1 are satisfied with ˜ 0 , k0 )−2 e2iν log(2k0 r = R(k0 )T(k

√

48k0 )

(5.22)

1 and ν = − 2π log(1 − |R(z0 )|2 ) since |r| = |R(z0 )|. Therefore we can conclude that the solution on c (k0 ) is given by

0 −β + O(t−α ) β 0  1 i 0 −β =I+ √ + O(t−α ), 48k0 (k − k0 ) t1/2 β 0 1 i ζ t1/2

M1c (k) = I +



(5.23)

where β is given by √ β = νei(π/4−arg(r)+arg((iν))) e−t(k0 ) t−iν √ ˜ 0 , k0 )2 (192k3 )−iν e−t(k0 ) t−iν = νei(π/4−arg(R(k0 ))+arg((iν))) T(k 0

(5.24)

and 1/2 < α < 1. We also need the solution M2c (k) on c (−k0 ). We make the following ansatz, which is inspired by the symmetry condition for the vector Riemann–Hilbert problem, outside the two small crosses: 

M2c (k)

 0 1 0 1 c = M1 (−k) 1 0 1 0  1 i 0 β =I− √ + O(t−α ). 48k0 (k + k0 ) t1/2 −β 0

(5.25)

Applying Theorem 5.2 yields the following result: Theorem 5.4 Assume  R

(1 + |x|)6 |q(x, 0)|dx < ∞,

(5.26)

Long-Time Asymptotics for the KdV Equation

313

then the asymptotics in the similarity region, x/t  −C for some C > 0, are given by  ∞   1 k0 q(y, t)dy = − 4 κj − log(|T(ζ )|2 )dζ π −k0 x κ j ∈(κ0 ,∞)

−





ν(k0 ) cos 16tk30 − ν(k0 ) log 192tk30 + δ(k0 ) + O(t−α ) 3k0 t (5.27)

respectively 



4ν(k0 )k0 sin 16tk30 − ν(k0 ) log 192tk30 + δ(k0 ) + O(t−α ) (5.28) 3t  x and for any 1/2 < α < 1. Here k0 = − 12t q(x, t) =

ν(k0 ) = −

1 log(|T(k0 )|), π

(5.29)

 N  κj π δ(k0 ) = − arg(R(k0 )) + arg((iν(k0 ))) + 4 arctan 4 k 0 j=1 1 − π





k0

log −k0

|T(ζ )|2 |T(k0 )|

2



1 dζ. ζ − k0

(5.30)

Proof By Theorem 5.2 we have 



i 1 1 1 (k) = 1 1 + √ m β −β − −β β + O(t−α ) k + k0 48k0 t1/2 k − k0 ∞   ∞ 





1 k0 l i 1  k0 l = 1 1 +√ − β −β − −β β k k 48k0 t1/2 k l=0

+ O(t

−α

l=0

),

which leads to 4 1 Q(x, t) = 2T1 (k0 ) + √ (Re(β)) + O(t−α ) 1/2 48k0 t  √  upon comparison with (4.38). Using the fact that β/ ν  = 1 proves the first claim. To see the second part, as in the proof of Theorem 4.4, just use (2.20) in place of (2.21), which shows  4k0 Im(β) + O(t−α ). q(x, t) = 3t

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K. Grunert, G. Teschl

This finishes the proof in the case where R(k) has an analytic extensions. We will remove this assumption in Section 6 thereby completing the proof.

Equivalence of the formula for δ(k0 ) given in the previous theorem with the one given in the introduction follows after a simple integration by parts. Remark 5.5 Formally the equation (5.28) for q can be obtained by differentiating the equation (5.27) for Q with respect to x. That this step is admissible could be shown as in Deift and Zhou [11], however our approach avoids this step. Remark 5.6 Note that Theorem 5.2 does not require uniform boundedness of the associated integral operator in contradistinction to Theorem A.6. We only need the knowledge of the solution in some small neighborhoods. However it cannot be used in the soliton region, because our solution is not of the form I + o(1).

6 Analytic Approximation In this section we want to present the necessary changes in the case where the reflection coefficient does not have an analytic extension. The idea is to use an analytic approximation and to split the reflection in an analytic part plus a small rest. The analytic part will be moved to the complex plane while the rest remains on the real axis. This needs to be done in such a way that the rest is of O(t−1 ) and the growth of the analytic part can be controlled by the decay of the phase. In the soliton region a straightforward splitting based on the Fourier transform  ˆ eikx R(x)dx (6.1) R(k) = R

will be sufficient. It is well-known that our decay assumption (4.32) implies Rˆ ∈ L1 (R) and the estimate (cf. [27, Sect. 3.2])  ∞ ˆ | R(−2x)|  const q(r)dr, x  0, (6.2) x

ˆ ∈ L1 (0, ∞). implies xl R(−x) ˆ Lemma 6.1 Suppose Rˆ ∈ L1 (R), xl R(−x) ∈ L1 (0, ∞) and let ε, β > 0 be given. Then we can split the reflection coefficient according to R(k) = Ra,t (k) + Rr,t (k) such that Ra,t (k) is analytic in 0 < Im(k) < ε and |Ra,t (k)e−βt | = O(t−l ),

0 < Im(k) < ε,

|Rr,t (k)| = O(t−l ),

k ∈ R. (6.3)

Long-Time Asymptotics for the KdV Equation

315

∞ ˆ Proof We choose Ra,t (k) = −K(t) eikx R(x)dx with K(t) = βε0 t for some positive β0 < β. Then, for 0 < Im(k) < ε,  ∞   − Im(k)x  Ra,t (k)e−βt   e−βt ˆ ˆ 1 =  R ˆ 1 e−(β−β0 )t , | R(x)|e dx  e−βt e K(t)ε  R −K(t)

which proves the first claim. Similarly, for Im(k) = 0,  ∞ l ˆ ˆ xl R(−x) x | R(−x)| const L1 (0,∞) dx   |Rr,t (k)| = l l x K(t) tl K(t)

To apply this lemma in the soliton region k0 ∈ iR+ we choose β = min − Re((k)) > 0.

(6.4)

Im(k)=ε

and split R(k) = Ra,t (k) + Rr,t (k) according to Lemma 6.1 to obtain b˜ ± (k) = b˜ a,t,± (k)b˜ r,t,± (k) = b˜ r,t,± (k)b˜ a,t,± (k).

(6.5)

Here b˜ a,t,± (k), b˜ r,t,± (k) denote the matrices obtained from b˜ ± (k) as defined in (4.26) by replacing R(k) with Ra,t (k), Rr,t (k), respectively. Now we can move the analytic parts into the complex plane as in Section 4 while leaving the rest on the real axis. Hence, rather then (4.28), the jump now reads ⎧ ˜ ⎪ k ∈ + , ⎨b a,t,+ (k), −1 ˜ v(k) ˆ = b a,t,− (k) , (6.6) k ∈ − , ⎪ ⎩˜ b r,t,− (k)−1 b˜ r,t,+ (k), k ∈ R. By construction we have v(k) ˆ = I + O(t−l ) on the whole contour and the rest follows as in Section 4. In the similarity region not only b˜ ± occur as jump matrices but also B˜ ± . These matrices B˜ ± have at first sight more complicated off diagonal entries, but a closer look shows that they have indeed the same form. To remedy this we will rewrite them in terms of left rather then right scattering data. For this purpose let us use the notation Rr (k) ≡ R+ (k) for the right and Rl (k) ≡ R− (k) for the left reflection coefficient. Moreover, let Tr (k, k0 ) ≡ T(k, k0 ) be the right and Tl (k, k0 ) ≡ T(k)/T(k, k0 ) be the left partial transmission coefficient. With this notation we have  Re(k0 ) < |k| , b˜ − (k)−1 b˜ + (k), (6.7) v(k) ˜ = −1 Re(k0 ) > |k| , B˜ − (k) B˜ + (k), where b˜ − (k) =

⎛ ⎝1 0

⎞ Rr (−k)e−t(k) Tr (−k, k0 )2 ⎠ , 1

⎛

1 b˜ + (k) = ⎝ Rr (k)et(k) Tr (k, k0 )2

⎞ 0 ⎠, 1

316

and

K. Grunert, G. Teschl

⎛

⎞ 1 0 ⎠, B˜ − (k) = ⎝ Tr,− (k, k0 )−2 t(k) − R (k)e 1 r |T(k)|2 ⎛ ⎞ Tr,+ (k, k0 )2 −t(k) 1 − R (−k)e r ⎠. B˜ + (k) = ⎝ |T(k)|2 0 1

Using (2.11) we can further write ⎞ ⎛ 1 0 ⎠, B˜ − (k) = ⎝ Rl (−k)et(k) 1 2 Tl (−k, k0 )

⎛ B˜ + (k) =

⎝1 0

⎞ Rl (k)e−t(k) Tl (k, k0 )2 ⎠ . 1

(6.8)

Now we can proceed as before with B˜ ± (k) as with b˜ ± (k) by splitting Rl (k) rather than Rr (k). In the similarity region we need to take the small vicinities of the stationary phase points into account. Since the phase is cubic near these points, we cannot use it to dominate the exponential growth of the analytic part away from the unit circle. Hence we will take the phase as a new variable and use the Fourier transform with respect to this new variable. Since this change of coordinates is singular near the stationary phase points, there is a price we have to pay, namely, requiring additional smoothness for R(k). In this respect note that (4.32) implies R(k) ∈ Cl (R) (cf. [23]). We begin with Lemma 6.2 Suppose R(k) ∈ C5 (R). Then we can split R(k) according to R(k) = R0 (k) + (k − k0 )(k + k0 )H(k),

k ∈ (k0 ),

(6.9)

where R0 (k) is a real rational function in k such that H(k) vanishes at k0 , −k0 of order three and has a Fourier transform  x(k) ˆ H(x)e dx, (6.10) H(k) = R

ˆ with x H(x) integrable. Proof We can construct a rational function, which satisfies fn (−k) = fn (k) k2n+4 +1 ! k 1 for k ∈ R, by making the ansatz fn (k) = k02n+4 +1 nj=0 j!(2k j (α j + iβ j k )(k − 0 0) k0 ) j(k + k0 ) j. Furthermore we can choose α j, β j ∈ R for j = 1, . . . , n, such that we can match the values of R and its first four derivatives at k0 , −k0 at fn (k). Thus we will set R0 (k) = f4 (k) with α0 = Re(R(k0 )), β0 = Im(R(k0 )), and so on. Since R0 (k) is integrable we infer that H(k) ∈ C4 (R) and it vanishes together with its first three derivatives at k0 , −k0 . Note that (k)/i = 8(k3 − 3k20 k) is a polynomial of order three which has a maximum at −k0 and a minimum at k0 . Thus the phase (k)/i restricted to

(k0 ) gives a one to one coordinate transform (k0 ) → [(k0 )/i, (−k0 )/i] = [−16k30 , 16k30 ] and we can hence express H(k) in this new coordinate (setting

Long-Time Asymptotics for the KdV Equation

317

it equal to zero outside this interval). The coordinate transform locally looks like a cube root near k0 and −k0 , however, due to our assumption that H vanishes there, H is still C2 in this new coordinate and the Fourier transform with respect to this new coordinates exists and has the required properties.

Moreover, as in Lemma 6.1 we obtain: Lemma 6.3 Let H(k) be as in the previous lemma. Then we can split H(k) according to H(k) = Ha,t (k) + Hr,t (k) such that Ha,t (k) is analytic in the region Re((k)) < 0 and |Ha,t (k)e(k)t/2 | = O(1),

Re((k)) < 0,

−1

|Hr,t (k)| = O(t ), Proof We choose Ha,t (k) = conclude as in Lemma 6.1:

∞

−K(t)

Im(k)  0,

k ∈ R.

(6.11)

x(k) ˆ H(x)e dx with K(t) = t/2. Then we can

−K(t)(k)+(k)t/2 ˆ ˆ |Ha,t (k)e(k)t/2 |   H(x) |   H(x) 1 |e 1  const

and  |Hr,t (k)| 

−K(t) −∞

ˆ | H(x)|dx  const



−K(t)

−∞

1 1 1 dx  const  const . x4 K(t)3/2 t



By construction Ra,t (k) = R0 (k) + (k − k0 )(k + k0 )Ha,t (k) will satisfy the required Lipschitz estimate in a vicinity of the stationary phase points (uniformly in t) and all jumps will be I + O(t−1 ). Hence we can proceed as in Section 5. Acknowledgements We want to thank Ira Egorova and Helge Krüger for helpful discussions and the referees for valuable hints with respect to the literature.

Appendix A: Singular Integral Equations In this section we show how to transform a meromorphic vector Riemann– Hilbert problem with simple poles at iκ, −iκ, m+ (k) = m− (k)v(k), k ∈ ,   0 0 0 −iγ 2 , Res , Resiκ m(k) = lim m(k) m(k) = lim m(k) −iκ 0 0 iγ 2 0 k→iκ k→−iκ  0 1 m(−k) = m(k) , 1 0

lim m(i k) = 1 1 (A.1)

k→∞

318

K. Grunert, G. Teschl

into a singular integral equation. Since we require the symmetry condition for our Riemann–Hilbert problem we need to adapt the usual Cauchy kernel to preserve this symmetry. Moreover, we keep the single soliton as an inhomogeneous term which will play the role of the leading asymptotics in our applications. The classical Cauchy-transform of a function f : → C which is square integrable is the analytic function C f : C\ → C given by C f (k) =

1 2π i



f (s) ds, s−k

k ∈ C\ .

(A.2)

Denote the tangential boundary values from both sides (taken possibly in the L2 -sense—see e.g. [8, Eq. (7.2)]) by C+ f respectively C− f . Then it is wellknown that C+ and C− are bounded operators L2 ( ) → L2 ( ), which satisfy C+ − C− = I (see e.g. [8]). Moreover, one has the Plemelj–Sokhotsky formula [29] C± =

1 (iH ± I), 2

where

 f (s) 1 H f (k) = − ds, k ∈ , (A.3) π k−s  is the Hilbert transform and − denotes the principal value integral. In order to respect the symmetry condition we will restrict our attention to the set L2s ( ) of square integrable functions f : → C2 such that  0 1 f (−k) = f (k) . (A.4) 1 0 Clearly this will only be possible if we require our jump data to be symmetric as well: Hypothesis A.1 Suppose the jump data ( , v) satisfy the following assumptions:

(i) consist of a finite number of smooth oriented finite curves in C which intersect at most finitely many times with all intersections being transversal. (ii) The distance between and {iy|y  y0 } is positive for some y0 > 0 and ±iκ ∈ . (iii) is invariant under k → −k and is oriented such that under the mapping k → −k sequences converging from the positive sided to are mapped to sequences converging to the negative side. (iv) The jump matrix v is invertible and can be factorized according to v = −1 b −1 − b + = (I − w− ) (I + w+ ), where w± = ±(b ± − I) satisfy   0 1 0 1 w± (−k) = w∓ (k) , k ∈ . (A.5) 1 0 1 0

Long-Time Asymptotics for the KdV Equation

319

(v) The jump matrix satisfies w∞ = w+  L∞ ( ) + w−  L∞ ( ) < ∞, w2 = w+  L2 ( ) + w−  L2 ( ) < ∞. Next we introduce the Cauchy operator  1 f (s)κ (s, k) (C f )(k) = 2π i

(A.6)

(A.7)

acting on vector-valued functions f : → C2 . Here the Cauchy kernel is given by ⎛ ⎞ k + iκ 1 0 ⎜ ⎟ κ (s, k) = ⎝ s + iκ s − k k − iκ 1 ⎠ ds 0 s − iκ s − k ⎞ ⎛ 1 1 − 0 ⎟ ⎜ (A.8) = ⎝ s − k s + iκ 1 1 ⎠ ds, − 0 s − k s − iκ for some fixed iκ ∈ / . In the case κ = ∞ we set ⎛ ⎞ 1 0 ⎜ ⎟ ∞ (s, k) = ⎝ s − k 1 ⎠ ds. (A.9) 0 s−k and one easily checks the symmetry property:   0 1 0 1 κ (s, k) . (A.10) κ (−s, −k) = 1 0 1 0 The properties of C are summarized in the next lemma. Lemma A.2 Assume Hypothesis A.1. The Cauchy operator C has the properties, that the boundary values C± are bounded operators L2s ( ) → L2s ( ) which satisfy C+ − C− = I

(A.11)

and (C f )(−iκ) = (0

∗),

(C f )(iκ) = (∗

Furthermore, C restricts to L2s ( ), that is  0 1 (C f )(−k) = (C f )(k) , 1 0 for f ∈ L2s ( ) or L∞ s ( ) and we also have

0).

(A.12)

k ∈ C\

(A.13)



0 1 C± ( f w∓ )(−k) = C∓ ( f w± )(k) , 1 0

k ∈ .

(A.14)

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K. Grunert, G. Teschl

Proof Everything follows from (A.10) and the fact that C inherits all properties from the classical Cauchy operator.

We have thus obtained a Cauchy transform with the required properties. Following Sections 7 and 8 of [3], we can solve our Riemann–Hilbert problem using this Cauchy operator. Introduce the operator Cw : L2s ( ) → L2s ( ) by Cw f = C+ ( f w− ) + C− ( f w+ ),

f ∈ L2s ( ).

(A.15)

By our hypothesis (A.6) Cw is also well-defined as operator from L∞ s ( ) → L2s ( ) and we have Cw  L2s →L2s  constw∞

2  constw2 . respectively Cw  L∞ (A.16) s →Ls

Furthermore recall from Lemma 2.6 that the unique solution corresponding to v ≡ I is given by

m0 (k) = f (k) f (−k) ,  1 k + iκ −1 2 t(iκ) f (k) = (2κ) 1 + γ e . 1 + (2κ)−1 γ 2 et(iκ) k − iκ Observe that for γ = 0 we have f (k) = 1 and for γ = ∞ we have f (k) = In particular, f (k) is uniformly bounded for all γ ∈ [0, ∞] if |k − iκ| > ε. Then we have the next result. Theorem A.3 Assume Hypothesis A.1. Suppose m solves the Riemann–Hilbert problem (A.1). Then  1 m(k) = (1 − c0 )m0 (k) + μ(s)(w+ (s) + w− (s))κ (s, k), 2π i

where −1 μ = m+ b −1 + = m− b −

 and c0 =

1 2π i



k+iκ . k−iκ

(A.17)

μ(s)(w+ (s) + w− (s))κ (s, ∞) . 1

Here (m) j denotes the j’th component of a vector. Furthermore, μ solves (I − Cw )(μ(k) − (1 − c0 )m0 (k)) = (1 − c0 )Cw m0 (k)

(A.18)

Conversely, suppose μ˜ solves ˜ − m0 (k)) = Cw m0 (k), (I − Cw )(μ(k) and

 c˜0 =

1 2π i



(A.19)

μ(s)(w ˜ (s) + w (s)) (s, ∞)

= −1, + − κ 1

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then m defined via (A.17), with (1 − c0 ) = (1 + c˜0 )−1 and μ = (1 + c˜0 )−1 μ, ˜ solves the Riemann–Hilbert problem (A.1) and μ = m± b −1 . ± Proof If m solves (A.1) and we set μ = m± b −1 ± , then m satisfies an additive jump given by m+ − m− = μ(w+ + w− ). ˜ both functions satisfy the Hence, if we denote the left hand side of (A.17) by m, same additive jump. Furthermore, Hypothesis 3.1 implies that μ is symmetric ˜ Using (A.12) we also see that m ˜ satisfies the same pole and hence so is m. ˜ has no jump and solves (A.1) with v ≡ I conditions as m0 . In summary, m − m ˜ except for the normalization which is given by limk→∞ m(ik) − m(ik) = (0 0). ˜ = 0. Hence Lemma 2.6 implies m − m Moreover, if m is given by (A.17), then (A.11) implies m± = (1 − c0 )m0 + C± (μw− ) + C± (μw+ ) = (1 − c0 )m0 + Cw (μ) ± μw± = (1 − c0 )m0 − (I − Cw )μ + μb ± .

(A.20)

From this we conclude that μ = m± b −1 ± solves (A.18). Conversely, if μ˜ solves (A.19), then set  1 ˜ μ(s)(w ˜ m(k) = m0 (k) + + (s) + w− (s))ζ (s, k), 2π i

˜ ± = μb and the same calculation as in (A.20) implies m ˜ ± , which shows that ˜ solves the Riemann–Hilbert problem (A.1). m = (1 + c˜0 )−1 m

Remark A.4 In our case m0 (k) ∈ L∞ s ( ), but m0 (k) is not square integrable ( ) in general. and so μ ∈ L2s ( ) + L∞ s Note also that in the special case γ = 0 we have m0 (k) = 1 1 and we can choose κ as we please, say κ = ∞ such that c0 = c˜0 = 0 in the above theorem. Hence we have a formula for the solution of our Riemann–Hilbert problem m(k) in terms of m0 + (I − Cw )−1 Cw m0 and this clearly raises the question of bounded invertibility of I − Cw as a map from L2s ( ) → L2s ( ). This follows from Fredholm theory (cf. e.g. [37]): Lemma A.5 Assume Hypothesis A.1. The operator I − Cw is Fredholm of index zero, ind(I − Cw ) = 0.

(A.21)

By the Fredholm alternative, it follows that to show the bounded invertibility of I − Cw we only need to show that ker(I − Cw ) = 0.

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We are interested in comparing a Riemann–Hilbert problem for which w∞ and w2 is small with the one-soliton problem. For such a situation we have the following result: Theorem A.6 Fix a contour and choose κ, γ = γ t , v t depending on some parameter t ∈ R such that Hypothesis A.1 holds. Assume that w t satisfies w t ∞  ρ(t) and wt 2  ρ(t)

(A.22)

for some function ρ(t) → 0 as t → ∞. Then (I − Cwt )−1 : L2s ( ) → L2s ( ) exists for sufficiently large t and the solution m(k) of the Riemann–Hilbert problem (A.1) differs from the one-soliton solution mt0 (k) only by O(ρ(t)), where the error term depends on the distance of k to ∪ {±iκ}. Proof By (A.16) we conclude that Cwt  L2s →L2s = O(ρ(t))

2 = O(ρ(t)) respectively Cwt  L∞ s →Ls

Thus, by the Neumann series, we infer that (I − Cwt )−1 exists for sufficiently large t and (I − Cwt )−1 − I L2s →L2s = O(ρ(t)). Next we observe that μ˜ t − mt0 = (I − Cwt )−1 Cwt mt0 ∈ L2s ( ) implying μ˜ t − mt0  L2s = O(ρ(t))

and c˜t0 = O(ρ(t))

since mt0 ∞ = O(1) (note μ˜ t0 = μt0 = mt0 ). Consequently ct0 = O(ρ(t)) and thus mt (k) − mt0 (k) = O(ρ(t)) uniformly in k as long as it stays a positive distance away from ∪ {±iκ}.



References 1. Ablowitz, M.J., Newell, A.C.: The decay of the continuous spectrum for solutions of the Korteweg–de Vries equation. J. Math. Phys. 14, 1277–1284 (1973) 2. Ablowitz, M.J., Segur, H.: Asymptotic solutions of the Korteweg–de Vries equation. Stud. Appl. Math. 57, 13–44 (1977) 3. Beals, R., Coifman, R.: Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37, 39–90 (1984) 4. Beals, R., Deift, P., Tomei, C.: Direct and inverse scattering on the real line. In: Mathematical Surveys and Monographs, vol. 28. American Mathematical Society, Providence (1988) 5. Budylin, A.M., Buslaev, V.S.: Quasiclassical integral equations and the asymptotic behavior of solutions of the Korteweg–de Vries equation for large time values. Dokl. Akad. Nauk 348(4), 455–458 (1996) (in Russian) 6. Buslaev, V.S.: Use of the determinant representation of solutions of the Korteweg–de Vries equation for the investigation of their asymptotic behavior for large times. Uspekhi Mat. Nauk 36(4), 217–218 (1981) (in Russian)

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Math Phys Anal Geom (2009) 12:325–380 DOI 10.1007/s11040-009-9065-z

Gauge Theory on Infinite Connected Sum and Mean Dimension Masaki Tsukamoto

Received: 10 February 2009 / Accepted: 30 July 2009 / Published online: 7 August 2009 © Springer Science + Business Media B.V. 2009

Abstract We study the geometry of infinite dimensional moduli spaces in the Yang-Mills gauge theory over infinite connected sum spaces. We develop the technique of gluing infinitely many instantons, and apply it to the evaluation of the mean dimension of the moduli spaces. Keywords Yang-Mills gauge theory · Mean dimension · Infinite connected sum · Gluing infinitely many instantons Mathematics Subject Classifications (2000) 58D27 · 53C07

1 Introduction Nonlinear analysis on non-compact manifolds is a challenging research field. We study the infinite energy Yang-Mills gauge theory on certain non-compact 4-manifolds (infinite connected sums of S4 ). Let  be a finitely generated infinite group with a finite generating set S. We suppose that S does not contain the identity element. We consider the infinite connected sum space (S4 )(,S) by gluing the copies of S4 ‘along the Cayley graph of (, S)’. (Its precise definition will be given in Section 2.) (S4 )(,S) is a

This work was supported by Grant-in-Aid for JSPS Fellows (19·1530) from Japan Society for the Promotion of Science. M. Tsukamoto (B) Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan e-mail: [email protected]

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non-compact 4-manifold.  naturally acts on (S4 )(,S) . For example, if (, S) = (Z, {1}), then (S4 )(,S) is conformally equivalent to S3 × R. Fix c  0. We want to study SU(2)-ASD connections A on (S4 )(,S) satisfying ||F A || L∞  c. (Here we consider ‘L∞ -norm condition’ for simplicity of the explanation. We will consider more general conditions later.) Since the base space (S4 )(,S) is non-compact, such ASD connections can have infinite L2 energy, and their moduli space M can be an infinite dimensional space. The moduli space M admits a natural -action. The main subject of this paper is the evaluation of the ‘mean dimension’ dim(M : ). Mean dimension is a notion introduced by Gromov [8]. (See also Lindenstrauss and Weiss [10] and Lindenstrauss [9].) Intuitively, the mean dimension dim(M : ) is given by dim(M : ) = dim M/||. (We give the precise definition of mean dimension in Appendix B.) In particular, if M is a finite dimensional space (in the usual sense), then dim(M : ) = 0. Hence the value of dim(M : ) has an information about ‘infinite dimensional geometry’ of M. We study M by using the technique of ‘gluing an infinite number of instantons’. Gluing instantons is a famous technique in the gauge theory. (Taubes [13], Donaldson [3], etc.) In Tsukamoto [14], we studied the technique of gluing infinitely many instantons. In the present paper we will develop this gluing technique more thoroughly and apply it to the evaluation of the mean dimension dim(M : ). The main body of the paper is devoted to the detailed (rather technical) study of this infinite gluing construction. The application of the gluing technique to the theory of mean dimension is suggested by Gromov [8, p. 403, 3.6.6] in the context of (pseudo-)holomorphic curves. Gournay [7] studies the application of the gluing technique of pseudoholomorphic curves to the problem of mean dimension. The motivations of this work come from several directions. The first motivation comes from the Nevanlinna theory (and the function theory over non-compact regions). There exists a (widely accepted, I believe) idea that there are many similarities between the Yang-Mills gauge theory and the theory of (pseudo-)holomorphic curves on their techniques and philosophies. (Donaldson’s invariant vs. Gromov-Witten’s invariant, etc.) At the holomorphic curve side, many researchers have developed the systematic studies of holomorphic maps from the complex plane C (and the complex space Cn ) since Nevanlinna [12] discovered his celebrated value distribution theory of meromorphic functions. In particular, they have studied holomorphic curves of infinite energy. The technique of ‘gluing an infinite number of instantons’ is the analogy of the Mittag-Leffler theorem in the classical function theory, and I believe that this paper is one-step toward the systematic study of infinite energy ASD connections and their infinite dimensional moduli spaces. (We studied the mean dimension of the moduli space of ‘Brody curves’ (cf. [2]) in [15, 16]. ASD connections satisfying the condition ||F(A)|| L∞  c are the

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analogy of Brody curves.) In this paper, we study only the mean dimension of the moduli space. So, of course, we need much more studies if we want to understand the infinite dimensional moduli spaces in the Yang-Mills theory. But dimension is one of the most fundamental quantitative invariant of spaces, and the techniques in this paper may be useful in the future studies. The second motivation comes from the study of non-linear analysis over ‘very big spaces’. The researchers of the geometric group theory (and the Novikov conjecture-type problems) have developed the study of analysis over non-compact (and ‘very big’) spaces. In this paper we study the gauge theory over (S4 )(,S) , and this space can be of infinite topological type. For example, if (, S) = (Z2 , {(1, 0), (0, 1)}), then the space (S4 )(,S) has infinitely many (linearly independent) cycles. Moreover, if  has exponential growth, then the space (S4 )(,S) also has exponential growth. I think that it is interesting (and surprising, I believe) that we can prove non-trivial results on the gauge theory over such big spaces. The third motivation comes from the theory of dynamical systems. One of the important tools in the study of dynamical systems is symbolic dynamics. Symbolic dynamics itself seems artificial at first sight, but the important point is that we can find it as a sub-dynamical system in many more natural dynamical systems (e.g. the study of homoclinic points). The study of the moduli space by the gluing technique can be seen as an attempt to find a symbolic dynamics (in a wider sense) inside an infinite dimensional dynamical system. Indeed, we will prove that, under some conditions, all ASD connections in M can be constructed from ‘gluing data’. The set of gluing data (and the group action on it) is the analogy of symbolic dynamics. For the dynamical systems perspective, see also Angenent [1].

2 Gauge Theory on Infinite Connected Sum of S 4 2.1 Main Results Let  be a finitely generated infinite group. Let S = {s1 , · · · , s|S| } ⊂  be a finite generating set which does not contain the identity element. Here we don’t suppose that S = S−1 . Easy examples are (, S) = (Z, {1}), (Z2 , {(1, 0), (0, 1)}). Let S4 be the 4-sphere and xs and ys (s ∈ S) be 2|S| distinct points in S4 . We will construct an infinite connected sum (S4 )(,S) by patching the copies of S4 ‘along the Cayley graph of (, S)’. The following construction is based on the ‘conformal connected sum’ described in Donaldson and Kronheimer [5, Section 7.2]. Since the standard metric on S4 is conformally flat, S4 has a Riemannian metric h satisfying the following: (i) h is conformally equivalent to the standard metric. (ii) h is flat in some neighborhood of each xs and ys (s ∈ S).

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Of course, h is not uniquely determined by these conditions. The important condition is the first one. The second condition is just for simplicity. Let λ be a positive (very small) parameters. For x ∈ S4 and r > 0, we denote B(x, r) (resp. ¯ B(x, r)) as the open (resp. closed) ball of radius r centered at x (with respect to the metric h). Set     √ √    U := S4 \ B¯ xs , λ/2  B¯ ys , λ/2 . s∈S

For each s ∈ S we choose an orientation-reversing linear isometry σs : Txs S4 → T ys S4 .   For each γ ∈ , let S4γ , hγ be the isometric copy of (S4 , h). Let xγ ,s and yγ ,s (s ∈ S) be the points on S4γ corresponding to xs and ys on S4 . S4γ has the open set U γ corresponding to U in S4 . We define the connected sum (S4 )(,S) by ⎛ ⎞  (S4 )(,S) := ⎝ U γ ⎠ / ∼, γ ∈

where the identification ∼√ is given as follows; We identify √  √  the annulus √ re-     gion B xγ ,s , 2 λ \ B¯ xγ ,s , λ/2 in S4γ with B yγ s,s , 2 λ \ B¯ yγ s,s , λ/2 in S4γ s by √  √ √  √       B xγ ,s , 2 λ \ B¯ xγ ,s , λ/2 ξ ∼ η ∈ B yγ s,s , 2 λ \ B¯ yγ s,s , λ/2 , def

⇐⇒ η = λσs (ξ )/|ξ |2 .

(1)

Here ξ and η are the normal coordinates centered at xγ ,s and yγ s,s , and we consider σs as a map from Txγ ,s S4γ to T yγ s,s S4γ s by identifying Txγ ,s S4γ (resp. T yγ s,s S4γ s ) with Txs S4 (resp. T ys S4 ). (S4 )(,S) admits a natural left -action as follows. For δ ∈  we define δ : U γ → U δγ by sending p ∈ U γ to q ∈ U δγ corresponding to the same point in S4 . This is compatible with the above identification (1). This action is fixed point free, i.e., every δ = 1 in  has no fixed point. We choose a -invariant Riemannian metric g on (S4 )(,S) as follows; Let 4 4 w be a smooth  function in the  in√S such that√0   w  1 all over S , w =  1√ complement of s∈S B xs , λ  B ys , λ , and w = 0 on each B¯ xs , λ/2     √ and B¯ ys , λ/2 . Let wγ γ ∈  be the copy of w defined in S4γ . We set g :=



w γ hγ ,

γ ∈

where hγ is the Riemannian metric given before. Since the map η = λσs (ξ )/|ξ |2 in (1) is conformal, g is conformally equivalent to each hγ over U γ .

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We want to study SU(2)-ASD connections over (S4 )(,S) . Let c  0 be a non-negative real number and d ∈ (2, +∞] (d may be +∞). Let E be a principal SU(2)-bundle over (S4 )(,S) and A be an ASD connection on E. We want to study such a pair (E, A) satisfying ||F A || Ld (Uγ ,g)  c

for all γ ∈ ,

(2)

where the norm denotes the Ld -norm over the region U γ defined by the metric g. Let (E, A) and (E , A ) be two pairs of a principal SU(2)-bundle over (S4 )(,S) and an ASD connection on it. They are called gauge equivalent if there exists a bundle isomorphism u : E → E satisfying u(A) = A . We define M(c, d ) as the space of the gauge equivalence classes [E, A] of a principal SU(2)-bundle E over (S4 )(,S) and an ASD connection A on E satisfying (2). This space admits a natural right -action: For [E, A] ∈ M(c, d ) and γ ∈ , we set

 [E, A].γ := γ ∗ E, γ ∗ A ,

(3)

where γ ∗ E and γ ∗ A are the pull-backs of E and A by the map γ : (S4 )(,S) → (S4 )(,S) . Remark 2.1 Since (S4 )(,S) is a non-compact 4-manifold, all principal SU(2)bundle on it are isomorphic to the product bundle (S4 )(,S) × SU(2). Therefore we can define M(c, d ) as the space of gauge equivalence classes of ASD connections on (S4 )(,S) × SU(2) satisfying (2). But the above formulation is more flexible. Remark 2.2 An ASD connection A satisfying the condition (2) is a YangMills analogue of ‘Brody curve’ in the theory of entire holomorphic curves. (cf. Brody [2], Tsukamoto [15, 16].) A holomorphic curve f : C → C P N is called a Brody curve if it satisfies |df |(z)  1 (or |df |(z)  C for some positive constant C) for all z ∈ C. M(c, d ) is equipped with the topology of C ∞ -convergence on compact subsets. That is, a sequence {[En , An ]}n1 ⊂ M(c, d ) converges to [E, A] ∈ M(c, d ) if for any compact set K ⊂ (S4 )(,S) there exist n0 (K) > 0 and bundle maps un : En | K → E| K (for n  n0 (K)) such that un (An | K ) converge to A| K in the C ∞ -topology. This topology is metrizable. From d > 2 and Uhlenbeck’s compactness result [17, Theorem 1.5 (3.6)], [18], the moduli space M(c, d ) becomes compact. But I think that this compactness is not so obvious. In some cases (e.g. (, S) = (Z, {1})) it directly follows from [18, Theorem E’]. But the general case needs some clarification. So we will give its proof in Appendix A.

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The group action M(c, d ) ×  → M(c, d ) defined in (3) is continuous. If  is amenable, then we can define the mean dimension dim(M(c, d ) : ). (See Gromov [8], Lindenstrauss and Weiss [10], Lindenstrauss [9] and Appendix B.) For c  0 and d > 2 we define MS4 (c, d ) as the space of the gauge equivalence classes of SU(2)-ASD connections [A] on S4 satisfying ||F A || Ld  S4 ,h  c. We denote dim MS4 (c, d ) as the topological (covering) dimension of MS4 (c, d ). For d ∈ (2, +∞] (d may be +∞), we set   c0 (d ) := sup c  0| dim MS4 (c, d ) = 0 ,   = sup c  0| MS4 (c, d ) = {[the product connection]} . We have c0 (d ) > 0, and dim MS4 (c, d ) > 0 for any c > c0 (d ). Our main results on the gauge theory over (S4 )(,S) are the following. The first result concerns with the upper bound on the mean dimension: Theorem 2.3 (i) For any d ∈ (2, +∞] and 0  c < c0 (d ) there exists λ0 (c, d ) > 0 such that if λ  λ0 (c, d ) then M(c, d ) is equal to the space of the gauge equivalence classes of flat SU(2)-connections on (S4 )(,S) . Hence if (S4 )(,S) is simply connected, then M(c, d ) is the one-point space. (ii) Suppose  is amenable. Then for any 0  c < c < +∞ and d ∈ (2, +∞], there exists λ1 (c, c, d ) > 0 such that if λ  λ1 (c, c, d ) then dim(M(c, d ) : )  3|S| + dim MS4 (c, d ). The next result is the lower bound on the mean dimension. Theorem 2.4 Suppose  is amenable. Let 0 < c < c < +∞ and 2 < d < +∞ (d must be finite). If dim MS4 (c, d ) > 0, then there exists λ2 (c, c, d ) > 0 such that for any λ  λ2 (c, c, d ) dim(M(c, d ) : )  3|S| + dim MS4 (c, d ). Theorem 2.5 Suppose  is amenable. For each d ∈ (2, +∞) there exists a countable set (d ) ⊂ (c0 (d ), +∞) satisfying the following: For any c ∈ (c0 (d ), +∞) \

(d ) there exists λ3 (c, d ) > 0 such that if λ  λ3 (c, d ) then we have dim(M(c, d ) : ) = 3|S| + dim MS4 (c, d ). We will prove Theorems 2.3 and 2.4 in Sections 8 and 9. Here we prove Theorem 2.5, assuming Theorems 2.3 and 2.4. Proof of Theorem 2.5 Consider the following non-decreasing function (c0 (d ), +∞) → Z>0 ,

c → dim MS4 (c, d ).

(4)

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Let (d ) ⊂ (c0 (d ), +∞) be the set of points where (4) is not continuous. (d ) is a countable set. For any c ∈ (c0 (d ), +∞) \ (d ), if we choose c and c (c < c < c) sufficiently close to c, then we have dim MS4 (c, d ) = dim MS4 (c, d ) = dim MS4 (c, d ) > 0. Using Theorems 2.3 and 2.4, we get dim(M(c, d ) : ) = 3|S| + dim MS4 (c, d ), for λ  1.

 

2.2 Outline of the Proofs of the Main Theorems The proofs of the main theorems (Theorems 2.3, 2.4) need lengthy technical arguments. So we want to describe the brief outline of the proofs in this subsection. For c  0 and d ∈ (2, +∞], we call θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S a (c, d )-gluing data if the following conditions are satisfied: Each Eγ (γ ∈ ) is a principal SU(2)-bundle over S4γ , and Aγ is an ASD connection on Eγ satisfying    F(Aγ ) d 4  c. ργ ,s is an SU(2)-isomorphism from (Eγ )xγ ,s to (Eγ s ) yγ s,s . L (Sγ ,hγ ) We can consider a natural equivalence relation in the set of (c, d )-gluing data. For each (c, d )-gluing data θ, we will construct a principal SU(2)-bundle E(θ) over (S4 )(,S) and an ASD connection A(θ ) on it by using a ‘gluing construction’. The proof of Theorem 2.3 proceeds as follows; 0  c < c¯ < +∞ and d ∈ ¯ d )-gluing data θ sat(2, +∞]. For each [E, A] ∈ M(c, d ) we can find a (c, isfying [E, A] = [E(θ), A(θ)] (λ  1). (This is the most difficult part of the proof.) If c¯ < c0 (d ), then we can (easily) prove that A(θ ) is flat. This shows Theorem 2.3 (i). In general, we have ¯ d )-gluing data : ), dim(M(c, d ) : )  dim(space of (c, and the right-hand-side can be estimated as follows. (The following argument ¯ d )-gluing data. For each is not rigorous.) Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a (c, ¯ d ) parameters of deformation, and ργ ,s has γ ∈ , (Eγ , Aγ ) has dim MS4 (c, three parameters (for each s ∈ S). Therefore the number of ‘deformation ¯ d ) + 3|S|)||. Hence we have parameters’ of θ is (dim MS4 (c, ¯ d )-gluing data : ) = dim(space of (c, ¯ d )-gluing data)/||, dim(space of (c, ¯ d ) + 3|S|. ≈ dim MS4 (c,

(5)

From this we get Theorem 2.3 (ii). On the other hand, if 0 < c < c < ∞ and d ∈ (2, +∞) then we can prove that, for each (c, d )-gluing data θ, [E(θ), A(θ)] belongs to M(c, d ). (The proof of this facts needs d < ∞.) Therefore we have dim(M(c, d ) : )  dim(space of (c, d )-gluing data : ).

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Using (5), we get Theorem 2.4. The above argument does not explain the meaning of the condition ‘dim MS4 (c, d ) > 0’ in the assumption in Theorem 2.4. This condition concerns with the validity of the equation (5).

3 Infinite Gluing Construction: Preparations From this section we will develop a theory of ‘gluing infinitely many instantons’ for general closed 4-manifolds. Let X be a compact, oriented Riemannian 4-manifold with prescribed 2|S|-points xs and ys (s ∈ S). We suppose that the metric is flat in some neighborhood of each xs and ys . Fix a real number p with 2 < p < 4 and define q ∈ (4, +∞) by 1 − 4/ p = −4/q, p i.e., L1 → Lq . (These p and q are fixed throughout the paper.) 3.1 Infinite Connected Sum First we briefly describe a construction of an infinite connected sum of X. This is essentially the same as in Section 2. But we need to introduce one more extra parameter N > 0 for several technical reasons. Let λ and √ N be positive parameters. We choose them so that λ  1, N  1 and N λ  1. We set (we follow the notations of Donaldson and Kronheimer [5, Section 7.2]) 

   √ √    B¯ xs , λ/N  B¯ ys , λ/N , X := X \ s∈S



   √   √  B¯ xs , λ/2  B¯ ys , λ/2 . X := X \

s∈S

X

corresponds to the region U in Section regions √ 2. √ annulus  We define  (xs ) and (ys ) in X by (xs ) := B xs , N λ \ B¯ xs , λ/N and (ys ) := √   √   B ys , N λ \ B¯ ys , λ/N . For each s ∈ S we choose an orientationreversing isometry σs : Txs X → T ys X. Let Xγ (γ ∈ ) be the copy of X with the points xγ ,s and yγ ,s (s ∈ S) corresponding to xs and ys . Xγ has the open sets Xγ , Xγ

, (xγ ,s ), (yγ ,s ) corresponding to X , X

, (xs ), (ys ) in X, respectively. We define X (,S) by ⎛ X (,S) := ⎝



⎞ Xγ ⎠ / ∼,

γ ∈

where we identify (xγ ,s ) in Xγ with (yγ s,s ) in Xγ s by def

(xγ ,s ) ξ ∼ η ∈ (yγ s,s ) ⇐⇒ η = λσs (ξ )/|ξ |2 .

(6)

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Here ξ and η are the normal coordinates centered at xγ ,s and yγ s,s , and we consider σs as a map from Txγ ,s Xγ to T yγ s,s Xγ s as in Section 2.  freely acts

on X (,S) ; for g ∈ , we define g : Xγ → Xgγ by sending p ∈ Xγ to q ∈ Xgγ corresponding to the same point in X. Let gγ (γ ∈ ) be the Riemannian metric on Xγ which is the copy of the metric on X. Let w be  a smooth √ function in √ X satisfying 0  w  1, B(x , λ)  B(y , λ), and w = 0 on each w = 1√in the complement of s s s∈S √ B(xs , λ/2) and B(ys , λ/2). We define a metric on X (,S) by g :=



wγ g γ ,

γ ∈

where the weight function wγ is the copy of w. We have X (,S) =

 γ ∈

Xγ =



Xγ

,

γ ∈

and the Riemannian structure on X (,S) is independent of N. Hence the above connected sum construction is compatible with that in Section 2. The Riemannian metric g is conformally equivalent to gγ over Xγ (g = 2 mγ gγ ) and satisfies 1  mγ  N 2 on Xγ ,

1  mγ  23 on Xγ

.

(7)

Moreover, on each neighborhood of xγ ,s and yγ ,s , N 5/3  mγ  N 2

√  √ λ/N  |ξ |  λ/N 5/6 ,

(8)

where ξ is the Euclidean coordinate (in Xγ ) around xγ ,s or yγ ,s . The important point for the later argument is the following (essentially the same things are discussed in [5, pp. 293–294]): For a 1-form α and a 2-form ξ on Xγ we have q |α|qg dvolg = m4−q γ |α|gγ dvolgγ ,

p |ξ |gp dvolg = m4−2 |ξ |gpγ dvolgγ , γ

(9)

where dvolg denotes the volume form defined by g. Since 2 < p < 4 and q > 4, (7) implies ||α|| Lq (Xγ , g)  ||α|| Lq (Xγ , gγ )  N 2−8/q ||α|| L p (Xγ ,g) , ||α|| Lq (Xγ

, gγ )  81−4/q ||α|| Lq (Xγ

, g) , ||ξ || L p (Xγ , g)  ||ξ || L p (Xγ , gγ )  N 4−8/ p ||ξ || L p (Xγ ,g) , ||ξ || L p (Xγ

, gγ )  82−4/ p ||ξ || L p (Xγ

, g) .

(10)

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3.2 Gluing Principal SU(2)-Bundles Let M be a set of (not necessarily all) gauge equivalence classes of (E, A), where E is a principal SU(2)-bundle over X and A is an ASD connection on E. We suppose that M can be decomposed as M = M0  M1 (disjoint union) satisfying the following conditions: M0 and M1 are compact with respect to the topology of C ∞ -convergence. (In particular, the number of the possible topological types of E is finite.) (b) For all [E, A] ∈ M0 , A is a regular connection. That is, the following two conditions are satisfied: A is irreducible (i.e., if a gauge transformation g : E → E satisfies g(A) = A, then g = ±1) and the operator d+A : 1 (adE) → + (adE) is surjective. Here 1 is the space of 1-forms and + is the space of self-dual 2-forms. adE is the Lie algebra bundle associated with E. (c) If X satisfies b1 (X) = b + (X) = 0 (e.g., X = S4 , C P2 ), then   M1 ⊂ [X × SU(2), the product connection] . (a)

Otherwise we set M1 = ∅. Therefore M1 is the one-point space or empty. Remark 3.1 Let E := X × SU(2) and A be the product connection. If b + (X) = 0, then d+A : 1 (adE) → + (adE) is surjective. But A is not irreducible. All constant gauge transformations fix A. The condition b + (X) = b1 (X) = 0 implies that [A] has no local deformation as an ASD connection. In our application to Theorem 2.3, we need to consider the product connection. The condition (c) is added for this purpose. But if the reader does not want to consider reducible connections, you should consider only the case M = M0 . Definition 3.2 A sequence (Eγ , Aγ , ργ ,s )γ ∈,s∈S is called a gluing data (or M-gluing data) if it satisfies the following: (i) For all γ ∈ , Eγ is a principal SU(2)-bundle over Xγ and Aγ is an ASD connection on it which satisfies [Eγ , Aγ ] ∈ M. (Here we naturally identify Xγ with X.) (ii) ργ ,s : (Eγ )xγ ,s → (Eγ s ) yγ s,s (γ ∈ , s ∈ S) is an SU(2)-isomorphism between the fibers (Eγ )xγ ,s and (Eγ s ) yγ s,s . We call ρ = (ργ ,s )γ ∈,s∈S ‘gluing parameter’. We consider that two gluing data (E1γ , A1γ , ρ1γ ,s )γ ∈,s∈S and (E2γ , A2γ , ρ2γ ,s )γ ∈,s∈S are equivalent if there exist bundle isomorphisms gγ : E1γ → E2γ (γ ∈ ) satisfying gγ (A1γ ) = A2γ and gγ s ρ1γ ,s = ρ2γ ,s gγ (in particular, E1γ and E2γ are isomorphic). We define GlD as the set of the equivalence classes of gluing data. (We sometimes use the notation ‘GlD M ’ when we need to make the dependence on M explicit.) There exists a natural projection GlD → M defined by (Eγ , Aγ , ργ ,s )γ ∈,s∈S → (Eγ , Aγ )γ ∈ .

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Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a gluing data. For x ∈ Xγ in a small neighborhood of xγ ,s , the fiber of Eγ over the point x can be identified with the fiber over xγ ,s by using the parallel transport (defined by Aγ ) along the radial line from xγ ,s to x. This trivialization is usually called ‘exponential gauge’ (or sometimes ‘radial gauge’); see [6, Chapter 9] or [5, Section 2.3.1]. Using these exponential gauges centered at xγ ,s or yγ ,s , we trivialize the bundle Eγ over s∈S (xγ ,s )  (yγ ,s ). ((xγ ,s ) and (yγ√ regions over ,s ) are the annulus √ ¯ γ ,s , λ/N).) Xγ defined in Section 3.1: (xγ ,s ) = B(xγ ,s , N λ) \ B(x Each ργ ,s is an isomorphism between (Eγ )xγ ,s and (Eγ s ) yγ s,s . We have the identification (xγ ,s ) ∼ = (yγ s,s ) defined by (6), and the above exponential gauges give the bundle trivializations Eγ |(xγ ,s ) ∼ = (xγ ,s ) × (Eγ )xγ ,s and Eγ s |(yγ s,s ) ∼ = (yγ s,s ) × (Eγ s ) yγ s,s . Therefore ργ ,s gives an identification map between Eγ |(xγ ,s ) and Eγ s |(yγ s,s ) covering the base space identification (xγ ,s ) ∼ = (yγ s,s ) (see the diagram (11)). exp. gauge

Eγ |(xγ ,s ) −−−−−→ (xγ ,s ) × (Eγ )xγ ,s ⏐ ⏐ ⏐ ⏐ργ ,s  

(11)

exp. gauge

Eγ s |(yγ s,s ) −−−−−→ (yγ s,s ) × (Eγ s ) yγ s,s We define a principal SU(2) bundle E(θ) over X (,S) by setting ⎛ E(θ) := ⎝



⎞ Eγ | Xγ ⎠ / ∼,

γ ∈

where we identify Eγ |(xγ ,s ) with Eγ s |(yγ s,s ) by (11). 3.3 Cut-Off Functions We need to introduce several cut-off functions. We basically follow the description of Donaldson and Kronheimer [5, Section 7.2]. First note that the following fact. Since M is compact, there exists an uniform upper bound on |F A | for all [E, A] ∈ M: |F A |  const M ,

(12)

where const M denotes a positive constant depending only on M. √ Set b :=  4N λ ( 1). Let ψ be the cut-off function on X such that ψ  = 0 over s∈S B(xs , b /2)  B(ys , b /2) and ψ = 1 over the complement of s∈S B(xs , b )  B(ys , b ) and |dψ|  4/b . Let ψγ be the copy of ψ defined on Xγ .

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Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a gluing data. As in Section 3.2, we trivialize the bundle Eγ around xγ ,s and yγ ,s by using the exponential gauges. Then we can define a connection A γ on Eγ by setting A γ := ψγ Aγ .

(13)

Here we consider Aγ as a connection matrix over each neighborhood of xγ ,s and yγ ,s by using the above trivialization. In the exponential gauge we have |Aγ (x)|  |x| sup |F Aγ |  const M |x| (see Donaldson and Kronheimer [5, p. 54]). Therefore we have   |A γ − Aγ |  const M · b , |F + A γ |  const M ,   |F A γ − F(Aγ )|  const M , (14) where const M is a positive constant which only depends on M (and is independent of γ , b , λ, N). Then    +   2  A − Aγ  4  F A  p  const · b ,  const M · b 4/ p , M γ γ L (Xγ ,gγ ) L (Xγ ,gγ )      F A − F(Aγ ) p  const M · b 4/ p . (15) γ L (X ,g ) γ

γ

A γ and A γ s (s ∈ S) coincide with each other over Xγ ∩ Xγ s under the identification (11). Hence there exists an unique (not necessarily ASD) connection A (θ) on E(θ) compatible with each A γ over Xγ . Remark 3.3 If [Eγ , Aγ ] ∈ M1 (i.e., Aγ is gauge equivalent to the product connection), then A γ = Aγ . Hence if [Eγ , Aγ ] ∈ M1 for all γ ∈ , then A (θ ) is a flat connection on E(θ) (which might have a non-trivial holonomy). Later (in Section 5.1) we will need the following {ψγ } also; Let ψ be the 

cut-off function on X such that  ψ = 0 over s∈S B(xs , b /4)  B(ys , b /4) and ψ = 1 over the complement of s∈S B(xs , b /2)  B(ys , b /2) and |dψ |  8/b . Let ψγ be the copy of ψ defined on Xγ . The following lemma is essentially the copy of [5, Lemma (7.2.10)]: Lemma 3.4 There exists a positive number K satisfying the following: For any λ and N there βλ,N defined in R4 such that β(x) = 0 √ exists a smooth function √β = 5/6 for |x|  λ/N, β(x) = 1 for |x|  λ/N and ||dβ|| L4  K(log N)−3/4 . Proof Note that the L4 -norm of a 1-form is conformally invariant. So we can 3 R4 to the cylinder × R by the change the description from the Euclidean √ √ √ S 5/6 coordinate transform t = log |x| − log λ. λ/N  |x|  λ/N becomes − log N  t  −(5/6) log N. Then the proof is easy.  

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√ √ The condition supp(dβ) ⊂ { λ/N  |x|  λ/N 5/6 } will be used in Section 6.1 (cf. (8)). Using the above Lemma 3.4, we define a cut-off function β on X by putting the above βλ,N around each xγ√and yγ . That is, √β is a function with  0  β  1 such that β = 0 on s∈S B(xs , λ/N)  B(ys , λ/N), β = 1 on the √ √  complement of s∈S B(xs , λ/N 5/6 )  B(ys , λ/N 5/6 ) and ||dβ|| L4  K(log N)−3/4 .

(16)

(Strictly speaking, the above constant K should be (2|S|)1/4 K. But for simplicity we use the abuse of notation.) Let βγ be the copy of β defined on Xγ .

We need to introduce one more √ β be a smooth √ function on cut-off. Let

X such that 0  β  1, β = 0 on s∈S B(xs , λ/2)  B(ys , λ/2), β = 1 on √ √ 

 B(y the complement of s∈S B(xs , 2 λ) s , 2 λ) (hence supp β ⊂ X ). We     can choose β so that the L4 -norm dβ  L4 is independent of λ (and N). Since √ √ N  1, we have λ/2  λ/N 5/6 and hence β · β = β .

(17)

Let βγ be the copy of β defined on Xγ (βγ · βγ = βγ ). Moreover we choose β so that these βγ becomes a partition of unity on X (,S) : βγ = 1. (18) γ ∈

In particular we have βγ + βγ s = 1 over (xγ ,s ) = (yγ s,s ). 3.4 Preliminary Estimates In this subsection we prepare several estimates. I think that they are essentially well-known. Therefore we omit most of the proofs. If some readers feel this subsection cumbersome, you should skip it and return to this subsection when it is used. 3.4.1 Right Inverse of d+A   ∗ Let [E, A] ∈ M, and set A := d+A d+A : + (adE) → + (adE), where  + ∗ d A : + (adE) → 1 (adE) is the formal adjoint of d+A . From the conditions (b) and (c) in the beginning of Section 3.2, there exists the inverse −1 A (see  ∗ + 1 also Remark 3.1). Set PA := d+A · −1 :  (adE) →  (adE). P becomes A A a right inverse of d+A : d+A PA = 1. Remember that 2 < p< 4, q > 4 and 1−4/ p= −4/q. We have the Sobolev p embedding: L1 (X) → Lq (X). Since M is compact, there exists a positive constant const M depending only on M (and independent of A) such that ||PA ξ || Lq  const M ||ξ || L p ,

||d A PA (ξ )|| L p  const M ||ξ || L p ,

for any [E, A] ∈ M and any ξ ∈ + (adE).

(19)

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3.4.2 The Cohomology H 1A Let [E, A] ∈ M and set   H 1A := ker d∗A + d+A : 1 (adE) → (0 ⊕ + )(adE) . (If [E, A] ∈ M1 , then H 1A = 0.) There exists δ M > 0 such that for any α ∈ H 1A with ||α|| Lq  δ M we have α˜ = α(A, ˜ α) ∈ 1 (adE) satisfying the following: d∗A α˜ = 0,

F + (A + α) ˜ = 0,

||α˜ − α|| Lq  const M ||α||2Lq . (We have α(A, ˜ 0) = 0.) Moreover

  ||α|| Lq  const M · d Lq [A], [A + α] ˜ .

(20)

Here, for connections A1 and A2 on E, we define the L -distance d Lq ([A1 ], [A2 ]) by   d Lq [A1 ], [A2 ] := inf ||A2 − g(A1 )|| Lq (X) . q

g:E→E

Lemma 3.5 There is δ M > 0 such that if an ASD connection B on E with [E, B] ∈ M satisfies d Lq ([A], [B])  δ M then there exists α ∈ H 1A with ||α|| Lq  δ M satisfying [B] = [A + α]. ˜ Lemma 3.6 If we choose δ M sufficiently small, then for any ξ ∈ + (adE) and α ∈ H 1A with ||α|| Lq  δ M , ||PA (ξ ) − PA+α˜ (ξ )|| Lq  const M ||α|| Lq ||ξ || L p . 3.4.3 Auxiliary Estimates ¯ s , ε) and For ε > 0 let Xε ⊂ X be the complement of the union of the balls B(x ¯ s , ε) (s ∈ S): B(y    ¯ s , ε) ∪ B(y ¯ s , ε) . B(x Xε := X \ s∈S

Lemma 3.7 There is ε M > 0 such that if ε  ε M then for any two [Ei , Ai ] ∈ M (i = 1, 2) we have the following: (1) If E1 is isomorphic to E2 , then d Lq ([A1 ], [A2 ])  const M · d Lq ([A1 | Xε ], [A2 | Xε ]), where d Lq ([A1 | Xε ], [A2 | Xε ]) is given by d Lq ([A1 | Xε ], [A2 | Xε ]) :=

inf

g:E1 | Xε →E2 | Xε

||A2 − g(A1 )|| Lq (Xε ) .

(2) If E1 is not isomorphic to E2 , then d Lq ([A1 | Xε ], [A2 | Xε ])  const M > 0.

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339

For [E, A] ∈ M we denote I A as the set of gauge transformations g : E → E satisfying g(A) = A. If [E, A] ∈ M0 , then I A = {±1}, and if [E, A] ∈ M1 , then IA ∼ = SU(2) (the set of constant gauge transformations). Lemma 3.8 There is ε M > 0 such that if ε  ε M then we have the following: Let [E, A] ∈ M and g : E| Xε → E| Xε be a bundle map over Xε . Then min ||g − h||C 0 (Xε )  const M ||d A g|| Lq (Xε ) .

h∈I A

3.4.4 Estimates About the Exponential Gauge Let D ⊂ R4 be a ball centered at the origin in the Euclidean space R4 , and E = D × SU(2) be a principal SU(2) bundle over D with smooth (not necessarily ASD) connections A1 and A2 . Let ui : E → D × E0 (i = 1, 2) be the exponential gauges associated with Ai centered at the origin. (E0 is the  fiber /∂r = of E at the origin.) We have ∂ui /∂r = ui Ai,r (r = |x|) and hence ∂ u1 u−1 2   u1 A1,r − A2,r u−1 . Therefore 2 |u1 (x) − u2 (x)|  |x| · ||A1 − A2 ||C 0 (B) .

(21)

Let Bi := ui (Ai ) be the connection matrices in the exponential gauge (i = 1, 2). Lemma 3.9 |B1 − B2 |  r ||F(A1 ) − F(A2 )||C 0 +

r2 ||A1 − A2 ||C 0 (||F(A1 )||C 0 + ||F(A2 )||C 0 ) . 2

Proof We have 

r

Bi,θ = 0

 F(Bi )rθ dr = 0

r

ui F(Ai )rθ ui−1 dr,

where r, θ denote the polar coordinate. (Of course, θ has three components.) Hence  r  −1 (u1 − u2 )F(A1 )rθ u−1 B1,θ − B2,θ = 1 + u2 (F(A1 )rθ − F(A2 )rθ ) u1 0

Then



r

|B1 − B2 | 

  −1 + u2 F(A2 )rθ u−1 dr. 1 − u2

|u1 − u2 |(|F(A1 )| + |F(A2 )|) + |F(A1 ) − F(A2 )|dr,

0



r2 ||A1 − A2 ||C 0 (||F(A1 )||C 0 + ||F(A2 )||C 0 ) + r ||F(A1 ) − F(A2 )||C 0 . 2  

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Let A = A(t) be a family of connections on E depending smoothly on the parameter t ∈ (−1, 1). Let u = u(t) : E → D × E0 be the exponential gauge of A centered at the origin, and set B = B(t) = u(A). Suppose that there exists a family of sections w = w(t) of adE such that u(t) = u(0)ew(t) and w(0) ≡ 0. Let ψ and φ be smooth functions on D satisfying 0  φ, ψ  1. Set A1 = A1 (t) := u−1 (ψ B) (we consider B as a connection matrix), and A2 = A2 (t) := eφw (A1 (t)). Lemma 3.10              ∂ A2              1 + r|dφ| + 3r2 ||F(A(0))||C 0  ∂ A   + r d A ∂ A   .   ∂t    ∂t   0 ∂t t=0 C 0 t=0 t=0 C Proof We can assume that A(0) is already a connection matrix in the exponential gauge. Then u(0) ≡ 1, u(t) = ew(t) and B(0) ≡ A(0). Let (r, θ ) be the polar coordinate. Set Ar = Ar (t) := A(t), ∂/∂r. We have Ar (0) ≡ 0 and ∂u/∂r = uAr . Hence ∂ ∂r



  ∂u  ∂ Ar  =u . ∂t t=0 ∂t t=0

Since u = 1 at the origin for all t, we have ∂u/∂t = 0 at the origin. Hence          ∂u        r  ∂ A   .  ∂t    ∂t   0 t=0 t=0 C

(22)

We have 

r

Bθ = 0



r

F(B)rθ dr =

uF(A)rθ u−1 dr.

0

Differentiating this equation and using the above (22), we get                  ∂ B     r2 ||F(A(0))||C 0  ∂ A   + r d A ∂ A   .   ∂t   0   ∂t   ∂t t=0 C 0 t=0 t=0 C

(23)

We have d A u = (A − B)u. Differentiating this (and using u(0) ≡ 1), we get  dA

   ∂u  ∂ A ∂ B  = . − ∂t t=0 ∂t ∂t t=0

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We have A1 = u−1 (ψ B) = u−1 du + ψu−1 Bu and A(0) = B(0). Then     ∂ A1  ∂u  ∂ B  = d + (ψ − 1)[A, ∂u/∂t]| + ψ , A t=0 ∂t t=0 ∂t t=0 ∂t t=0   ∂ B  ∂ A  + (ψ − 1) + (ψ − 1)[A, ∂u/∂t]|t=0 . = ∂t t=0 ∂t t=0 We have w(0) ≡ 0, ∂w/∂t|t=0 = ∂u/∂t|t=0 and A1 (0) = ψ B(0) = ψ A(0). Some calculation shows    ∂ A2  ∂u  ∂ A  = − dφ ⊗ + (1 − φ) ∂t t=0 ∂t t=0 ∂t t=0  ∂ B  + (ψ + φ − 1) + (ψ − 1)(1 − φ)[A, ∂u/∂t]|t=0 . ∂t t=0 We have |A(0)|  r ||F(A(0))||C 0 . Therefore             ∂ A2               1 + r|dφ| + 3r2 ||F(A(0))||C 0  ∂ A   + r d A ∂ A   .   ∂t    ∂t   0 ∂t t=0 C 0 t=0 t=0 C   Moreover suppose that A = A(t) is a family of ASD connections. We have F + (A1 ) = u−1 F + (ψ B)u and F + (ψ B) = (dψ ∧ B)+ + ψ(ψ − 1)(B ∧ B)+ . Using the inequalities (22) and (23), we get (at t = 0)           ∂ +   F (A1 )  2r|F + (ψ B)|  ∂ A  +  ∂ F + (ψ B) ,   ∂t  0  ∂t   ∂t C   |F + (ψ B)|  r|dψ| + r2 ||F A ||C 0 ||F A ||C 0 ,       √   ∂ +   F (ψ B)  r|dψ| + 2 2r2 ||F A ||C 0 r ||F A ||C 0  ∂ A    ∂t  ∂t 

C0

     ∂ A  + d A . ∂t C 0 (24)

4 Infinite Gluing: Basic Construction In the following three sections we will develop the technique of gluing an infinite number of ASD connections. Our approach is based on the method of Donaldson and Kronheimer [5, Section 7.2]. We also use the ideas of Angenent [1] and Macrì et al. [11] in the construction of the right inverse of d+A . A different approach using ‘alternating method’ in Donaldson [4] is developed in Tsukamoto [14]. Recall that 2 < p < 4, q > 4 and 1 − 4/ p = −4/q.

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4.1 Construction Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a gluing data. That is, Eγ (γ ∈ ) is a principal SU(2) bundle over Xγ , and Aγ is an ASD connection on Eγ satisfying [Eγ , Aγ ] ∈ M. ρ := (ργ ,s )γ ∈,s∈S is a gluing parameter. We have constructed the principal SU(2) bundle E = E(θ) on X (,S) . We want to construct an ASD connection on E by gluing the given ASD connections Aγ . Let α and ξ be adE-valued 1-form and self-dual 2-form on X (,S) respectively. We define BLq -norm (bounded Lq -norm) of α and BL p -norm of ξ by ||ξ || BL p := sup ||ξ || L p (Xγ

, g) .

||α|| BLq := sup ||α|| Lq (Xγ

, g) , γ ∈

(25)

γ ∈

Let BLq be the Banach space of all locally-Lq , adE-valued 1-forms whose BLq -norms are finite, and BL p be the Banach space of all locally-L p , adEvalued self-dual 2-forms whose BL p -norms are finite. This type of function space is used in Macrì et al. [11] for the study of self-dual vortices. It is also used in Gournay [7] for the study of gluing infinitely many pseudo-holomorphic curves. For ξ ∈ BL p we define an adE-valued 1-form Q(ξ ) = Qθ (ξ ) by Q(ξ ) :=



  βγ PAγ βγ ξ ,

(26)

γ ∈

where PAγ is the right inverse of d+Aγ defined in Section 3.4.1. The above infinite p sum is a locally finite sum, and Q(ξ ) becomes locally-L1 . Using (19) and (10), we have (note that suppβγ ⊂ Xγ

)     PA β ξ  q γ γ L

(

Xγ , g

)

       PAγ βγ ξ  Lq (X , g )  const M βγ ξ  L p (X γ

γ

 const M ||ξ || L p (Xγ

, gγ )  const M ||ξ || L p (Xγ

, g) .

γ , gγ )

, (27)

Therefore ||Q(ξ )|| BLq  const M · ||ξ || BL p .

(28)

Let A = A (θ) be the connection on X (,S) defined in Section 3.3. We have d+A Q(ξ ) =

γ ∈

   d+A γ βγ PAγ βγ ξ .

Since d+Aγ PAγ = 1 and βγ βγ = βγ (see (17)),       +  d+A γ βγ PAγ βγ ξ = dβγ ∧ PAγ βγ ξ +βγ d+A γ PAγ βγ ξ ,   +

   + = βγ ξ + dβγ ∧ PAγ βγ ξ +βγ A γ − Aγ ∧ PAγ βγ ξ .

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{βγ } is a partition of unity (see (18)). So d+A Q(ξ ) = ξ +



   +  + dβγ ∧ PAγ βγ ξ + βγ Aγ − Aγ ∧ PAγ βγ ξ .

γ ∈

γ ∈

(29) From Hölder’s inequality (L4 × Lq → L p ) and (27)     +    +    +  A γ − Aγ ∧ PAγ βγ ξ  p  p  dβγ ∧ PAγ βγ ξ L ( Xγ , g) L ( Xγ , g)       const M · dβγ  L4 ( Xγ , g) +  Aγ − A γ  L4 X , g ||ξ || L p ( Xγ

, g) . ( γ )   Note that the L4 -norm of a 1-form is conformally invariant. So dβγ  L4 (Xγ , g)       and  Aγ − A γ  L4 (X , g) are equal to dβγ  L4 (Xγ , gγ ) and  Aγ − A γ  L4 (X , g ) , γ √ γ γ and these are very small (see (15) and (16)) for N  1 and b = 4N λ  1. Then we get  +    d Q(ξ ) − ξ  p  const M (log N)−3/4 + b 2 ||ξ || BL p . A BL Thus Lemma 4.1 Set R := Rθ := d+A Q − 1 : BL p → BL p . For any ξ ∈ BL p   ||R(ξ )|| BL p  const M (log N)−3/4 + b 2 ||ξ || BL p . Hence there exist positive that √ constants N0 = N0 (M) and b 0 = b 0 (M) such −1 ||R|| if N  N and b = 4N λ  b then  1/2 and there exists (1 + R) = 0 0  n p p −1 (−R) : BL → BL . P := P := Q(1 + R) gives a right inverse θ n0 of d+A : d+A P(ξ ) = ξ, for any ξ ∈ BL p . From (28), for any ξ ∈ BL p ||P(ξ )|| BLq  const M · ||ξ || BL p .

(30)

We want to find an ASD connection of the form A + P(ξ ) (ξ ∈ BL p ). Since P is a right inverse of d+A , this is equivalent to solving the following equation for ξ ∈ BL p :  + ξ + P(ξ ) ∧ P(ξ ) = −F + ( A ). (31) From (7) we have vol(Xγ

, g)  84 vol(Xγ

, gγ )  84 vol(X). From this and q > 4, we have sup ||P(ξ )|| L4 (Xγ

,g)  const · ||P(ξ )|| BLq , γ ∈

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where this ‘const’ is a positive constant depending only on vol(X). Then, using Hölder’s inequality (L4 × Lq → L p ) and (27), we get   (P(ξ1 ) ∧ P(ξ1 ))+ − (P(ξ2 ) ∧ P(ξ2 ))+  p BL  const M (||ξ1 || BL p + ||ξ2 || BL p ) ||ξ1 − ξ2 || BL p .

Then we use the following lemma (this is [5, p. 289, Lemma (7.2.23)]). Lemma 4.2 Let B be a Banach space and k be a positive constant. Let S : B → B be a (not necessarily linear) map satisfying S(0) = 0 and ||S(x) − S(y)||  k(||x|| + ||y||) ||x − y||. Then for any y ∈ B with ||y||  1/(10k) there uniquely exists x ∈ B with ||x||  1/(5k) satisfying x + S(x) = y. Moreover x satisfies ||x||  ||y|| + 2k ||y||2 . Proof Set T(x) := y − S(x). It is easy to check that for ||x||  1/(5k) we have ||T(x)||  1/(5k) and for ||xi ||  1/(5k) (i = 1, 2) we have ||T(x1 ) − T(x2 )||  (2/5) ||x1 − x2 ||. Then the contraction mapping principle implies that there uniquely exists x with ||x||  1/(5k) satisfying T(x) = x. If x + S(x) = y and ||x||  1/(5k), then ||x||  ||y|| + k ||x||2  ||y|| + (1/5) ||x||. Hence ||x||  (5/4) ||y||. Therefore ||x − y||  k ||x||2  (25k/16) ||y||2  2k ||y||2 .   From (15) we have

 +   F ( A )

BL p

 const M · b 4/ p .

Hence we can solve the equation (31) if b  1. 1 Proposition 4.3 There are positive constants √ N0 = N0 (M), b 0 = b 0 (M), C1 = C1 (M) such that if N  N0 and b = 4N λ  b 0 then there exists ξ = ξ (θ ) ∈ BL p with ||ξ || BL p  C1 satisfying

F + ( A + P(ξ )) = 0

and

||ξ || BL p  const M · b 4/ p .

Moreover this ξ is unique, i.e., if η ∈ BL p with ||η|| BL p  C1 satisfies F + ( A + P(η)) = 0 then η = ξ . From (30), ||P(ξ )|| BLq  const M · b 4/ p . We will denote A(θ) := A + P(ξ ).

1 This

is an abuse of notation; these N0 and b 0 are not necessarily equal to the constants in Lemma 4.1.

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345

Remark 4.4 We assume 2 < p < 4. But the above construction argument is still true for p = 2. In particular, we have ||ξ || BL2 := sup ||ξ || L2 (Xγ

,g)  const M · b 2 , γ ∈

||P(ξ )|| BL4 := sup ||P(ξ )|| L4 (Xγ

,g)  const M · b 2 . γ ∈

Remark 4.5 If [Eγ , Aγ ] ∈ M1 for all γ ∈ , then A (θ ) is a flat connection (cf. Remark 3.3). In particular, F + ( A (θ)) = 0. Hence we have ξ (θ ) = 0 and A(θ) = A (θ). 4.2 Estimate on the Curvature We want to estimate the curvature of A = A(θ). From (15),      F( A ) p  sup  F(Aγ ) p + const M · b 4/ p . BL L (Xγ ,gγ ) γ ∈

(32)

For any ξ ∈ BL p we have d A Q(ξ ) =

     dβγ ∧ PAγ βγ ξ + βγ d Aγ PAγ βγ ξ γ

+ βγ



    A γ − Aγ ∧ PAγ βγ ξ .

We can estimate this as in the previous subsection by using (19) and get: ||d A Q(ξ )|| BL p  const M ||ξ || BL p . Since P = Q(1 + R)−1 , we have   ||d A P(ξ )|| BL p  const M (1 + R)−1 ξ  BL p  const M ||ξ || BL p . We have A = A + P(ξ ) and F( A) = F( A ) + d A P(ξ ) + P(ξ ) ∧ P(ξ ). Using ||ξ || BL p  const M b 4/ p , we get   ||F( A)|| BL p   F( A ) BL p + ||d A P(ξ )|| BL p + const ||P(ξ )||2BLq     F( A ) BL p + const M b 4/ p . Using (32) we get the conclusion: Proposition 4.6 The ASD connection A(θ) satisfies   ||F( A(θ))|| BL p  sup  F(Aγ ) L p (X ,g ) + const M · b 4/ p . γ γ γ ∈

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5 Infinite Gluing: Injectivity Problem Sections 5 and 6 are technical. Some readers should skip these sections and go to Section 7, and return to them when the results in these sections are used. The main result in Section 5 is Proposition 5.5, and the main result in Section 6 is Theorem 6.11. They will be used later. Some arguments in Sections 5.1, 6.2 and 6.3 (in particular, Corollary 6.8 and Lemma 6.10) will be also used later. 5.1 Variation Let θ := (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a gluing data. For each γ ∈ , let αγ ∈ H 1Aγ   with αγ  Lq  δ M (see Section 3.4.2), and A˜ γ := Aγ + α˜ γ be the ASD connection on Eγ given in Section 3.4.2. Set α := (αγ )γ ∈ . Let vγ ,s ∈ (adEγ )xγ ,s (γ ∈ , s ∈ S) with |vγ ,s |  Diam(SU(2)). Set ργ ,s := ργ ,s evγ ,s and v := (vγ ,s )γ ∈,s∈S . We define   ||α|| := sup αγ  Lq (X

γ ,gγ )

γ ∈

,

||v|| := sup |vγ ,s |. γ ∈,s∈S

Suppose [Eγ , A˜ γ ] ∈ M and set θ˜ := (Eγ , A˜ γ , ργ ,s )γ ∈,s∈S . We want to compare ˜ with A := A(θ). First we will construct a gauge transformation h A˜ := A(θ) ˜ to E = E(θ). from E˜ = E(θ) Let uγ : Eγ | B(xγ ,s ,b ) → B(xγ ,s , b )×(Eγ )xγ ,s and uγ : Eγ | B(yγ ,s ,b ) → B(yγ ,s , b )× (Eγ ) yγ ,s be the exponential gauges of Aγ around xγ ,s and yγ ,s (γ ∈ , s ∈ S). We also denote u˜ γ as the exponential gauges of A˜ γ around xγ ,s and yγ ,s (γ ∈ , s ∈ S). From (21) we have |uγ − u˜ γ |  const M · b ||α||  1. Hence there uniquely exists a section wγ of adEγ with |wγ |  const M · b ||α||  1 over  , b )  B(yγ ,s , b ) satisfying u˜ γ = uγ ewγ . We define a section vˆγ of s∈S B(xγ ,s adEγ over s∈S B(xγ ,s , b )  B(yγ ,s , b ) by setting ⎧ ⎨u−1 γ ◦ vγ ,s ◦ uγ   vˆγ := ⎩−u−1 ◦ ργ s−1 ,s ◦ vγ s−1 ,s ◦ ρ −1−1 ◦ uγ γ γ s ,s

on B(xγ ,s , b ) (s ∈ S), on B(yγ ,s , b ) (s ∈ S).

(33)

We define the gauge transformation hγ : Eγ → Eγ by  hγ :=



e(1−βγ )vˆγ e(1−ψγ )wγ

on B(xγ ,s , b ) and B(yγ ,s , b ) (s ∈ S),

1

otherwise,

(34)

where βγ and ψγ are the cut-off functions introduced in Section 3.3. (ψγ 

satisfies |dψγ |  8/b , ψγ =0 over s∈S B(xs , b /4)  B(ys , b /4) and ψγ = 1

over the complement of s∈S B(xs , b /2)  B(ys , b /2).) Since βγ + βγ s = 1

Gauge Theory and Mean Dimension

347

√ and ψγ = 0 over (xγ ,s ) = (yγ s,s ) (remember: (xγ ,s ) = B(xγ ,s , N λ) \ √ ¯ γ ,s , λ/N)), the diagram (35) becomes commutative. B(x hγ

Eγ |(xγ ,s ) −−−−→ Eγ |(xγ ,s ) ⏐ ⏐ ⏐ ⏐u−1 ◦ρ ◦u

˜γ  u˜ −1  γ s γ ,s γ γ s ◦ργ ,s ◦u

(35)

hγ s

Eγ s |(yγ s,s ) −−−−→ Eγ s |(yγ s,s ) Therefore {hγ } compatibly define the gauge transformation h = hv,α : E˜ → E. ˜ Set P := Pθ and Pv,α := h ◦ Pθ˜ ◦ h−1 : BL p → BLq . We set A˜ := A (θ) (see Section 3.3) and A v,α := h( A˜ )

(this is a connection on E).

(36)

Lemma 5.1 For any ξ ∈ BL p ,    Pv,α (ξ ) − P(ξ ) q  const M (||α|| + ||v||) ||ξ || BL p . BL Proof The proof is just a confirmation of the definitions. We have  −1 Pv,α = Qv,α d+A v,α Qv,α , where Qv,α (ξ ) =

γ ∈

  hγ · βγ P A˜ γ βγ h−1 γ ξ ,

 

−1

= (hγ −1)βγ P A˜ γ (βγ h−1 (ξ ))+β P (β (h −1)ξ )+β P (β ξ ) . ˜ ˜ γ γ γ γ γ γ Aγ Aγ γ ∈

(37) By using the definition (34) and Lemma 3.6, we get    Qv,α (ξ ) − Q(ξ ) q  const M (||α|| + ||v||) ||ξ || BL p . BL In a similar way (cf. (29)),   −1  +  + −1   d Qv,α  (ξ ) − d Q (ξ )

A  Av,α 

 const M (||α|| + ||v||) ||ξ || BL p . BL p

Then we get the above conclusion.

 

  Set ξv,α := h(ξ (θ˜ )) ∈ BL p (+ (adE)). We have ξv,α  BL p  const M · b 4/ p (see Proposition 4.3). Lemma 5.2

  ξv,α − ξ  p  const M · b 4/ p · (||α|| + b 2 ||v||). BL

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Proof We have (see (31) and Proposition 4.3) ξv,α + (Pv,α (ξv,α ) ∧ Pv,α (ξv,α ))+ = −F + ( A v,α ), ξ + (P(ξ ) ∧ P(ξ ))+ = −F + ( A ). By using (21), Lemma 3.9 and hγ = 1 in the support of F + ( A˜ γ ) (cf. Section 3.3), we have  +    +

 4/ p  F A ||α|| . v,α − F ( A ) BL p  const M · b From Proposition 4.3, Remark 4.4, Lemma 5.1 and Hölder’s inequality BL4 × BLq → BL p ,   (Pv,α (ξv,α ) ∧ Pv,α (ξv,α ))+ − (P(ξ ) ∧ P(ξ ))+  p , BL   2  2+4/ p    const M · b ξv,α − ξ BL p + const M · b (||α|| + ||v||). Hence     ξv,α − ξ  p  const M · b 2 ξv,α − ξ  p + const M · b 4/ p (||α|| + b 2 ||v||). BL BL Since b  1, we get the desired estimate. Corollary 5.3

 

   Pv,α (ξv,α ) − P(ξ ) q  const M · b 4/ p (||α|| + ||v||). BL

Set aγ := P(ξ )| Xγ

∈ 1Xγ

(adEγ ) and a˜ γ := Pθ˜ (ξθ˜ )| Xγ

= h−1 γ (Pv,α (ξv,α ))| Xγ

∈ ˜ X

= A˜ + a˜ γ . 1

(adEγ ) for each γ ∈ . We have A| X

= A + aγ and A| Xγ

Lemma 5.4

γ

γ

γ

γ

  sup aγ − a˜ γ  Lq (X

,gγ )  const M · b 4/ p (||α|| + ||v||). γ ∈

γ

Proof

  a˜ γ − aγ = h−1 γ − 1 Pv,α (ξv,α )hγ + Pv,α (ξv,α )(hγ − 1) + (Pv,α (ξv,α ) − P(ξ )).

From Proposition 4.3 and Corollary 5.3, we get the above estimate.

 

5.2 Injectivity Problem The purpose of this subsection is to prove the following. We follow the method of [4, 5]. Proposition 5.5 There exists √ N0 = N0 (M) > 0 and b 0 = b 0 (M) > 0 such that if N  N0 and b = 4N λ  b 0 then the following holds: Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S and θ = (Fγ , Bγ , ργ ,s )γ ∈,s∈S be two gluing data. Then A(θ ) is gauge equivalent to A(θ ) if and only if [θ] = [θ ] in GlD.

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349

Proof The ‘if’ part is a direct consequence of the definitions. So we will give the proof of the ‘only if’ part. We set A1 := A(θ ) and A2 := A(θ ). Suppose there is a gauge transformation g : E(θ) → E(θ ) satisfying g( A1 ) = A2 . We ¯ γ ,s , b ) and B(y ¯ γ ,s , b ) define Xγ ,b (γ ∈ ) as the complement of the b -balls B(x in Xγ : Xγ ,b := Xγ \

 

 ¯ γ ,s , b ) ∪ B(y ¯ γ ,s , b ) . B(x

s∈S

By the definitions of the cut-off functions in Section 3.3, we have A γ = Aγ and B γ = Bγ over Xγ ,b . From Proposition 4.3, we have      Aγ − A1  q ,  Bγ − A2  Lq (Xγ ,b ,gγ )  const M · b 4/ p . L (Xγ ,b ,gγ ) Since A1 is gauge equivalent to A2 ,   d Lq [Aγ | Xγ ,b ], [Bγ | Xγ ,b ]  const M · b 4/ p  1. From Lemma 3.7 (2), this implies (for b  1) Eγ ∼ = Fγ for all γ ∈ . Moreover, from Lemma 3.7 (1), Lemma 3.5 and the inequality (20), there exists   αγ ∈ H 1Aγ with αγ  Lq  const M · b 4/ p for each γ ∈  such that Bγ is gauge equivalent to A˜ γ := Aγ + α˜ γ . We can suppose Bγ = A˜ γ without loss of generality. ργ ,s and ργ ,s are SU(2)-isomorphisms between (Eγ )xγ ,s and (Eγ s ) yγ s,s (γ ∈ , s ∈ S). Take vγ ,s ∈ (adE)xγ ,s such that ργ ,s = ργ ,s evγ ,s and |vγ ,s | = d(ργ ,s , ργ ,s ) ( Diam(SU(2))). Set α := (αγ )γ ∈ and v := (vγ ,s )γ ∈,s∈S as in Section 5.1. From the assumption, there are gauge transformations gγ of Eγ over Xγ

such that   gγ A γ + aγ = A˜ γ + a˜ γ

over Xγ

,

(38)

where aγ and a˜ γ are the element of 1Xγ

(adEγ ) satisfying A(θ ) = A γ + aγ and A(θ ) = A˜ γ + a˜ γ over Xγ

as in Section 5.1. Moreover gγ satisfy the following compatibility condition: ργ ,s ◦ gγ = gγ s ◦ ργ ,s

over (xγ ,s ) = (yγ s,s ).

(39)

Let I Aγ be the isotropy group of Aγ . From Lemma 3.8, we have       min gγ − hC 0 (X

)  const M d A γ gγ  γ

h∈I Aγ

Using the action of

 γ ∈

Lq (Xγ

,gγ )

.

I Aγ on the gluing data, we can assume that

    gγ − 1 0

 const M d A gγ  γ C (X ) γ

Lq (Xγ

,gγ )

.

(40)

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From (38), d A γ gγ

Lq (Xγ

)

 A γ − A˜ γ

Lq (Xγ

)

Using Lemma 3.9, we get

      +2 gγ −1C 0 (X

) aγ  Lq (X

) + a˜ γ −aγ  Lq (X

) . γ

A γ − A˜ γ

Lq (Xγ

)

γ

γ

 const M · αγ  Lq (Xγ ) . From

Proposition 4.3 and Lemma 5.4, we get aγ  Lq (Xγ

)  const M · b 4/ p  1 and aγ − a˜ γ  Lq (Xγ

)  const M · b 4/ p (α + v). Therefore (using (40))   gγ − 1 0

 const M ||α|| + const M · b 4/ p ||v|| . (41) C (X ) γ

On the other hand, from (38), gγ (Aγ ) − A˜ γ = (˜aγ − aγ ) + (1 − gγ )aγ + gγ aγ (1 − gγ−1 ) over Xγ ,b where A γ = Aγ and A˜ γ = A˜ γ . Hence (using (20) and Lemma 3.7 (1))  "    ! αγ   const M · d Lq Aγ | X , A˜ γ | X γ ,b γ ,b         const M aγ − a˜ γ  Lq (X

) + gγ − 1C 0 (X

) aγ  Lq (Xγ ) . γ

γ

Using (41), Proposition 4.3 and Lemma 5.4, ||α||  const M · b 4/ p (||α|| + ||v||). Since b  1, we get ||α||  const M · b 4/ p ||v|| .   Substituting this into (41), we get gγ − 1C 0 (X

)  const M · b 4/ p ||v||.

(42)

γ

From the compatibility condition (39), ργ ,s − ργ ,s = (gγ s − 1)ργ ,s gγ + ργ ,s (gγ−1 − 1). Hence     |ργ ,s − ργ ,s |  gγ s − 1C 0 (X

) + gγ − 1C 0 (X

)  const M · b 4/ p ||v|| . γ

γ

Then ||v||  const · sup |ργ ,s − ργ ,s |  const M · b 4/ p ||v|| . γ ,s

Since b  1, we get ||v|| = 0. This implies ρ = ρ and (using (42)) ||α|| = 0. Therefore Bγ = A˜ γ = Aγ .  

6 Infinite Gluing: Surjectivity Problem In this section we will study a ‘surjectivity problem’. We basically follow the argument of Donaldson and Kronheimer [5, Sections 7.2.4, 7.2.5]. But our case is more involved because we cannot use the usual index-theorem argument. (This difficulty is suggested in [5, p. 298].)

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351

6.1 Linearized Problem Let θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S be a gluing data. Let I ⊂  be the set of γ ∈  satisfying [Eγ , Aγ ] ∈ M1 . If M1 = ∅, then I = ∅. Set E := E(θ ) and A := A (θ). We define V and H by ⎫ ⎧ ⎬ ⎨ # (adEγ )xγ ,s | ||v|| := sup |vγ ,s | < ∞ , V := v = (vγ ,s )γ ∈,s∈S ∈ ⎭ ⎩ γ ,s H :=

⎧ ⎨ ⎩

γ ∈,s∈S

α = (αγ )γ ∈ ∈

# γ ∈

⎫ ⎬     H 1Aγ | ||α|| := sup αγ  Lq (Xγ ,gγ ) < ∞ . ⎭ γ

(43)

Note that H 1Aγ = 0 for [Eγ , Aγ ] ∈ M1 . For v ∈ V and α ∈ H, we define j1 (v) and j2 (α) in 1 (adE) by ∂  ∂  j1 (v) :=  A tv,0 , j2 (α) :=  A 0,tα , (44) ∂t t=0 ∂t t=0 where A v,α is the connection on E defined as in (36). ργ ,s is a SU(2)-isomorphism between the fibers (Eγ )xγ ,s and (Eγ s ) yγ s,s . In this subsection, we identify the fibers (Eγ )xγ ,s and (Eγ s ) yγ ,s by the given ργ ,s . Let v = (vγ ,s )γ ,s where vγ ,s ∈ (adEγ )xγ ,s ∼ = (adEγ s ) yγ s,s . We often consider vγ ,s as a section of adEγ (or adEγ s ) over (xγ ,s ) (or (yγ s,s )) by using the exponential gauge of Aγ (or Aγ s ): Eγ |(xγ ,s ) ∼ = (xγ ,s ) × (Eγ )xγ ,s (or Eγ s |(yγ s,s ) ∼ = (yγ s,s ) × (Eγ s ) yγ s,s ). Then the above j1 (v) is expressed by    d A γ βγ vγ s = dβγ ⊗ vγ ,s over (xγ ,s ) (45) j1 (v) =   −d A γ βγ vγ s−1 ,s = −dβγ ⊗ vγ s−1 ,s over (yγ ,s ), ' ' and we have supp( j1 (v)) ⊂ γ supp(dβγ ) ⊂ γ ,s ((xγ ,s ) ∪ (yγ ,s )). From this we easily deduce that d+A j1 (v) = 0. Lemma 6.1

     

 j2 (α) −  β α γ γ     γ ∈

 const M · b 4/q ||α|| . BLq

In particular, || j2 (α)|| BLq  const M ||α||. Moreover  +  d j2 (α) p  const M · b 4/ p ||α|| . A BL Proof We have j2 (α) = αγ over Xγ \ 3.10, we have | j2 (α)|gγ  const M ||α||

'

s (B(xγ ,s , b )∪ B(yγ ,s , b )).

over

 s

Using Lemma

 B(xγ ,s , b ) ∪ B(yγ ,s , b ) .

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M. Tsukamoto

Therefore we get the first inequality. We have d+A j2 (α) = 0 over Xγ \ ' (B(xγ ,s , b )∪ B(yγ ,s , b )), where F + ( A 0,tα ) = 0. Since d+A j2 (α) = ∂ F + ( A 0,tα )/ s ∂tt=0 and hγ = 1 in the support of F + ( A 0,tα ), the estimate (24) gives  +  d j2 (α)  const M ||α|| A

over



 B(xγ ,s , b ) ∪ B(yγ ,s , b ) .

γ ,s

  Therefore d+A j2 (α) BL p  const M · b 4/ p ||α||.

 

√ Fix z ∈ X \ {xs , ys |s ∈ S}. We take b = 4N λ > 0 so small that the balls of radius b around xs and ys (s ∈ S) don’t contain z. Let zγ (γ ∈ ) be the point in Xγ corresponding to z. We define (adE)0 as the set of χ ∈ (adE) satisfying χ(zγ ) = 0 for all γ ∈ I (i.e., [Eγ , Aγ ] ∈ M1 ). Here we consider zγ ∈ Xγ

⊂ X (,S) . Lemma 6.2 For any χ ∈ 0 (adE)0 , v ∈ V and α ∈ H, ||χ||C 0 + ||v|| + ||α||  const M ||d A χ + j1 (v) + j2 (α)|| BLq . Proof For each γ ∈  we define a section χγ of adEγ over Xγ

by χγ := χ +

(βγ − 1)vγ ,s + (1 − βγ )vγ s−1 ,s . s

s

We have d A γ χγ = d A χ + j1 (v) over Xγ

and χγ s − χγ = vγ ,s over Xγ

∩ Xγ

s .   We have ||χ||C 0 + ||v||  (4|S| + 3) supγ χγ C 0 (X

) . If γ ∈ I, then χγ (zγ ) = 0. If γ γ ∈ / I, then Aγ is irreducible. Therefore         χγ  0

+ αγ  q  const M d A γ χγ + αγ  q

, C (X ) L (Xγ ) L (Xγ ,gγ )

γ

 const M ||d A χ + j1 (v) + j2 (α)|| BLq   + const M αγ − j2 (α) Lq (X

,gγ ) . γ

By using the argument in the proof of Lemma 6.1, we get   αγ − j2 (α) q

 const M · b 4/q ||α|| . L (X ,g ) γ

γ

Since b  1, we get the above conclusion.

 

Let χ ∈ 0 (adE)0 and ξ ∈ + (adE) be smooth (not necessarily compact supported) 0-form and self-dual 2-form valued in adE over X (,S) , and let v ∈ V and α ∈ H. We define the norm ||(χ, v, α, ξ )|| B1 by ||(χ, v, α, ξ )|| B1 := ||d A χ + j1 (v) + j2 (α)|| BLq + ||ξ || BL p .

(46)

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353

Lemma 6.2 shows that this becomes a norm. (Of course, its value might be infinity.) We define the Banach space B1 as the completion of the space of (χ, v, α, ξ ) ∈ 0 (adE)0 ⊕ V ⊕ H ⊕ + (adE) of ||(χ, v, α, ξ )|| B1 < ∞ in the norm ||·|| B1 :   B1 := (χ, v, α, ξ ) ∈ 0 (adE)0 ⊕ V ⊕ H ⊕ + (adE)| ||(χ, v, α, ξ )|| B1 < ∞ , where the overline means the completion in the norm ||·|| B1 . Let ω ∈ 1 (adE) be a smooth 1-form valued in adE over X (,S) . We define the norm ||ω|| B2 by setting   ||ω|| B2 := ||ω|| BLq + d+A ω BL p . (47) We define the Banach space B2 as the completion of the space of ω ∈ 1 (adE) of ||ω|| B2 < ∞ in the norm ||·|| B2 :   B2 := ω ∈ 1 (adE)| ||ω|| B2 < ∞ . Let P = Pθ : BL p → BLq be the map defined in Lemma 4.1. P is a right inverse of d+A . We define a linear map T : B1 → B2 by T(χ, v, α, ξ ) := d A χ + j1 (v) + j2 (α) + P(ξ ).

(48)

Proposition 6.3 T is a bounded linear operator. Moreover there exists a positive constant K depending only on M such that for any (χ, v, α, ξ ) ∈ B1 ||(χ, v, α, ξ )|| B1  K ||T(χ, v, α, ξ )|| B2 . Proof Set ω := T(χ, v, α, ξ ). We have (using d+A j1 (v) = 0)

 + d+A ω = F + A , χ + d A j2 (α) + ξ. From 6.1 and 6.2, T is bounded. From this equation (remember  + Lemmas   F ( A ) p  const M b 4/ p ) BL ||ξ || BL p  ||ω|| B2 + const M · b 4/ p (||χ||C 0 + ||α||),  ||ω|| B2 + const M · b 4/ p ||d A χ + j1 (v) + j2 (α)|| BLq ,

= ||ω|| B2 + const M · b 4/ p ||ω − P(ξ )|| BLq ,  (1 + const M · b 4/ p ) ||ω|| B2 + const M · b 4/ p ||ξ || BL p .

Since b  1, we get ||ξ || BL p  2 ||ω|| B2 and ||d A χ + j1 (v) + j2 (α)|| BL p = ||ω − P(ξ )|| BL p  const M ||ω|| B2 .

 

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This result shows that T is an embedding. Indeed we want to prove that T is an isomorphism. Donaldson and Kronheimer [5, Section 7.2.5] proves a similar result by using the index theorem. But we cannot use the index theorem and must prove it by a direct analysis.   Let ω ∈ B2 and set ω := ω− Pd+A ω. We have d+A ω = 0 and ω  BLq  const M ||ω|| B2 . Consider βγ ω on Xγ for each γ ∈ . We have d+Aγ {βγ ω − Pγ d+Aγ (βγ ω )} = 0 where Pγ = (d+Aγ )∗ (d+Aγ (d+Aγ )∗ )−1 is the right inverse of d+Aγ . Then there uniquely exist χγ ∈ 0 (adEγ ) and αγ ∈ H 1Aγ such that   d Aγ χγ + αγ = βγ ω − Pγ d+Aγ βγ ω , and χγ (zγ ) = 0 if γ ∈ I. (If γ ∈ I, then H 1Aγ = 0 and αγ = 0.) Since d+A ω = 0,    +

   + Pγ d+Aγ βγ ω = Pγ dβγ ∧ ω + Pγ Aγ − A γ ∧ βγ ω .

(49)

A difficulty comes from the term Pγ (dβγ ∧ ω )+ . The term Pγ [(Aγ − A γ ) ∧ (βγ ω)]+ can be easily estimated:     +    Pγ Aγ − A γ ∧ βγ ω 

Lq (Xγ ,gγ )

     const M  Aγ − A γ  L4 βγ ω  Lq (X ,gγ ) , γ

 const M · b 2 N 2−8/q ||ω|| B2 .

(50)

Here we have used (10) and (15). If we choose b 2 N 2−8/q  1, then this is a good estimate. But a similar estimation gives   +    const M · N 2−8/q ||ω|| B2 .  Pγ dβγ ∧ ω  q L (Xγ ,gγ )

Since N  1 and 2 − 8/q > 0, this is not a small term. We will come back to this point later. We have (χγ (zγ ) = 0 for γ ∈ )       χγ  0 + αγ  q  const M d Aγ χγ +αγ  Lq (Xγ ,gγ )  const M · N 2−8/q ||ω|| B2 . C L (Xγ ,gγ ) (51)  Set χ := γ βγ χγ ∈ 0 (adE)0 . Then we have the following equation (using βγ βγ = βγ ): d A χ −

γ

= ω + −



γ

dβγ ⊗ χγ +

γ

γ

βγ αγ



+  βγ Aγ − A γ , χγ − βγ Pγ (Aγ − A γ ) ∧ (βγ ω )

 + βγ Pγ dβγ ∧ ω .

(52)

Gauge Theory and Mean Dimension

355

From (51),        Aγ − A , χγ  q

  Aγ − A  q

  γ γ L (X ,g ) χγ C 0 L (X ,g) γ

γ

 const M · b

    +    Pγ Aγ − A γ ∧ βγ ω 

Lq (Xγ

,g)

1+4/q

N

γ

2−8/q

||ω|| B2 .

(53)

     const M  Aγ − A γ  L4 βγ ω Lq (X ,gγ ) , γ

 const M · b 2 N 2−8/q ||ω|| B2 .

(54)

Hence     

     



+  βγ Aγ − Aγ , χγ − βγ Pγ Aγ − Aγ ∧ βγ ω     γ

 const M · b

1+4/q

N

2−8/q

BLq

||ω|| B2 ,

 const M · b N 2 ||ω|| B2 .

The estimation of the term

 γ

(55)

βγ Pγ (dβγ ∧ ω )+ needs the following lemma:

Lemma 6.4 Let 0 < δ < 1, and f be a L p -function in R4 satisfying √  √  supp f ⊂ x| λ/N  |x|  λ/N 1−δ , where N  1. Set  F(x) :=

R4

f (y) dy. |x − y|3

Then we have 

1/q √ |x| λ/2

|F(x)|q dx

 const · N −4(1−1/ p)(1−δ) || f || L p (R4 ) .

Here remember that 2 < p < 4, q > 4 and 1 − 4/ p = −4/q. Proof Using a scale change, we suppose λ = 1 without loss of generality. If |x|  1/2 and |y|  N −1+δ , then (using N  1 and −1 + δ < 0) |x − y|  |x| − N −1+δ  |x|/2. Then 1/|x − y|3  23 /|x|3 . Hence for |x|  1/2 |F(x)| 

23 |x|3

 R4

| f (y)|dy.

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M. Tsukamoto

Using q > 4, we get 

 |F(x)| dx  2 q

|x|1/2

3q

∞

1/2





dr r3q−3

q R4

| f (y)|dy

1/q

|x|1/2

|F(x)|q dx

  const

q R4

| f (y)|dy

.

  const

R4

| f (y)|dy,

 const || f || L p (R4 ) (vol (supp f ))1−1/ p ,  const · N −4(1−δ)(1−1/ p) || f || L p (R4 ) .

  We use this lemma for δ = 1/6. Lemma 6.5    Pγ (dβγ ∧ ω )+  q

 const M · N −1/2 ||ω|| B2 . L (X ,gγ ) γ

Proof Set σ := (dβγ ∧ ω )+ and δ = 1/6. There exists r0 > 0 (independent of λ and N) such that the metric gγ is flat over the √balls Bs = B(xγ ,s , r0 ) ' and B s = B(yγ ,s , r0 ) (s ∈ S). We assume r0  b = 4N λ. Set B√:= (Bs ∪ A s by As√:= B(xγ ,s , λ/N 1−δ ) \ B s ). We√define the annulus regions √ As and

1−δ ¯ γ ,s , λ/N) and As := B(yγ ,s , λ/N ) \ B(y ¯ γ ,s , λ/N), and set A := B(x ' (As ∪ A s ). Remember that supp σ ⊂ supp(dβγ ) ⊂ A by Lemma 3.4. Pγ σ can be expressed by using the Green kernel:  Pγ σ (x) =

G(x, y)σ (y)dvol(y), A

where the volume form dvol(y) = dvolgγ (y) is defined by using the metric gγ . The Green kernel G(x, y) has a singularity of degree 3 along the diagonal (cf. Donaldson [4, p. 310]): |G(x, y)|  const M /d(x, y)3 , where d(x, y) is the distance on Xγ defined by gγ . 

 Xγ

|Pγ σ |q dvol =

 Xγ

\B

|Pγ σ |q dvol +

B∩Xγ

|Pγ σ |q dvol.

Gauge Theory and Mean Dimension

357

The first term can be easily estimated:  q   q |Pγ σ | dvol  const M dvol(x) |σ (y)|dvol(y) , Xγ

\B

Xγ

A



q/ p |σ (y)| dvol(y) ,

 const M · (volA)

q(1−1/ p)

p

A

 q(1−1/ p)  const M · λ2 N −4(1−δ)



q/ p |σ (y)| dvol(y) . p

A

From Lemma 6.4, 

1/q

|Pγ σ | dvol q

B∩Xγ

 const M · N −4(1−δ)(1−1/ p)



1/ p |σ (y)| p dvol(y) . A

Hence    Pγ σ 

Lq (Xγ

,gγ )

 const M · N −4(1−δ)(1−1/ p) ||σ || L p (Xγ ,gγ ) .

From (10),       ||σ || L p (Xγ ,gγ )  dβγ  L4 ω  Lq (X ,g )  const M · N 2−8/q ω  Lq (X ,g) γ γ

 const M · N

2−8/q

γ

||ω|| B2 .

We have 1 − 4/ p = −4/q, 2 < p < 4 and δ = 1/6. Then 2 − 8/q − 4(1 − δ)(1 − 1/ p) = 4δ(1 − 1/ p) − 4/ p < −1/2. Therefore we get the above conclusion.   From the above Lemma, we get     

+  βγ Pγ (dβγ ∧ ω )     γ

 const M · N −1/2 ||ω|| B2 . BLq

From the equation (52) and the estimate (55), we get       

 dβγ ⊗ χγ + βγ αγ − ω   const M bN 2 + N −1/2 ||ω|| B2 . d A χ −   γ

γ

BLq

(56) √ √ Let Wγ ,s := B(xγ ,s , 2 λ) \ B(xγ ,s , λ/2) ⊂ Xγbe the ‘neck’ region (γ ∈ , s ∈ S). Since dβγ = −dβγ s over Wγ ,s , the term − γ dβγ ⊗ χγ can be expressed by − dβγ ⊗ χγ = dβγ ⊗ (−χγ + χγ s )|Wγ ,s . γ

γ ,s

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M. Tsukamoto

We have d Aγ χγ = βγ ω − Pγ d+Aγ (βγ ω ) − αγ . Since βγ = 1 over the neck Wγ ,s , we have  

 d A χγ = ω − Pγ d+Aγ βγ ω − αγ + A γ − Aγ , χγ on Wγ ,s . Therefore on the neck Wγ ,s   

d A (χγ − χγ s ) = −Pγ d+Aγ βγ ω − αγ + A γ − Aγ , χγ   

+ Pγ s d+Aγ s βγ s ω + αγ s − A γ s − Aγ s , χγ s . As in (54) and Lemma 6.5,      Pγ d+Aγ (βγ ω ) q

L (Wγ ,s ,g)

(57)

   const M · N −1/2 + b2 N 2−8/q ||ω|| B2 .

From (53),    Aγ − A , χγ  q

 const M · b1+4/q N 2−8/q ||ω|| B . γ 2 L (X ,g) γ

    From (51) and an (elliptic) estimate αγ  L∞ (Xγ ,gγ )  const M αγ  Lq (Xγ ,gγ ) ,     αγ  q  (const · λ2 )1/q αγ  L∞ (Xγ ,gγ ) , L (Wγ ,s ,g)    const M · λ2/q αγ  Lq (Xγ ,gγ )  const M · λ2/q N 2−8/q ||ω|| B2 . In the same way, we get the estimates of the other terms in the right-hand-side of (57). Then     d A (χγ − χγ s ) q  const M · N −1/2 + bN 2 + λ2/q N 2 ||ω|| B2 . L (Wγ ,s ,g) Let vγ ,s ∈ (adEγ )xγ ,s ∼ = (adEγ s ) yγ s,s be the mean value of χγ s − χγ over the neck q Wγ ,s . Using the Sobolev embedding L1 → C 0,1−4/q (Hölder space), we get     χγ s − χγ − vγ ,s  0  const · λ1/2−2/q d A (χγ − χγ s ) Lq (Wγ ,s ,g) , C (Wγ ,s )    const M · λ1/2−2/q N −1/2 + bN 2 + λ2/q N 2 ||ω|| B2 . Set v := (vγ ,s )γ ,s ∈ V.      

dβγ ⊗ χγ + j1 (v)   γ 

BLq

     

=  dβγ ⊗ (vγ ,s − χγ s + χγ )|Wγ ,s   γ ,s 

, BLq

  1 √  const M · √ ( λ)4/q λ1/2−2/q N −1 + b N 2 + λ2/q N 2 ||ω|| B2 , λ  −1/2   const M N + bN 2 + λ2/q N 2 ||ω|| B2 .

(58)

Gauge Theory and Mean Dimension

359

  Set α := (αγ )γ ∈ H. From (51), ||α|| = sup αγ  Lq  const M · N 2−8/q ||ω|| B2 . √ Using Lemma 6.1, we get (b = 4N λ)      

 j2 (α) − βγ αγ   const M · b4/q ||α||  const M · b 4/q N 2−8/q ||ω|| B2 ,   q  γ ∈ BL

 const M · λ2/q N 2 ||ω|| B2 .

Using this and (58) in the estimate (56), we get     d A χ + j1 (v) + j2 (α) − ω  q  const M · N −1/2 + bN 2 + λ2/q N 2 ||ω|| B . 2 BL       +  +  We have d+A d A χ + j1 (v) + j2 (α) − ω = [F + A , χ] + d A j2 (α). Using F A BL p  4/ p const M · b , (51) and Lemma 6.1, we get  +    F , χ + d+ j2 (α) p  const M · b4/ p N 2−8/q ||ω|| B + const M · b4/ p ||α|| , A A 2 BL  const M · λ2/q N 2 ||ω|| B2 .

Thus we conclude that     d A χ + j1 (v) + j2 (α) − ω   const M · N −1/2 + bN 2 + λ2/q N 2 ||ω|| B . 2 B2 We define a bounded linear operator T : B2 → B1 byT (ω) := (χ,v, α, Remember ω = ω − Pd+A ω. The above shows TT (ω) − ω B2  const M · (N −1/2 + bN 2 + λ2/q N 2 ) ||ω|| B2 . Therefore if we choose λ and N appropriately, then (TT )−1 exists and T (TT )−1 becomes a right inverse of T. In particular, T becomes surjective and hence isomorphic (see Proposition 6.3). Then we get the following. d+A ω).

Proposition 6.6 There are N0 > 0 and λ0 (N) > 0 such that if N  N0 and λ  λ0 (N) then T : B1 → B2 (given in (48)) is an isomorphism and satisfies ||(χ, v, α, ξ )|| B1  K ||T(χ, v, α, ξ )|| B2 , where K is a positive constant depending only on M. 6.2 Some Continuities Let’s recall our situation.  is a finitely generated group and S is its finite generating set which does not contain the identity element e. The group  can be considered as a metric space endowed with the (left-invariant) word distance by S: For γ , γ ∈ ,   d S (γ , γ ) := min n  0| ∃γ1 , · · · , γn ∈ S ∪ S−1 : γ −1 γ = γ1 · · · γn For a subset  ⊂  and an integer d > 0, we set Bd () := {γ ∈ | ∃γ ∈  : d S (γ , γ )  d}.

360

M. Tsukamoto

We define a open set X ⊂ X by X :=



Xγ

.

γ ∈

Let θi = (Eiγ , Aiγ , ρiγ ,s )γ ∈,s∈S (i = 1, 2) be two gluing data, i.e., Eiγ is a principal SU(2)-bundle over Xγ and Aiγ is an ASD connection on Eiγ satisfying [Eiγ , Aiγ ] ∈ M. ρiγ ,s : (Eiγ )xγ ,s → (Eiγ s ) yγ s,s is an SU(2)-isomorphism. For each i = 1, 2, we have the operator Pi : BL p (+ (adEi )) → BLq (1 (adEi )) which is a right inverse of d+A by Lemma 4.1. Let  ⊂  be a finite set. We i want to compare the operators P1 and P2 over X . Suppose that there is an integer d > 0 such that E1γ = E2γ , A1γ = A2γ for γ ∈ Bd () and ρ1γ ,s = ρ2γ ,s for γ ∈ Bd () and s ∈ S with γ s ∈ Bd (). Then we can naturally consider that E1 = E2 and A 1 = A 2 over X Bd () . Lemma 6.7 Let ξi ∈ BL p (+ (adEi )) (i = 1, 2). We denote ξi | X Bd () as the restriction of ξi to X Bd () (and we extend it to X (,S) by zero). Then for each γ ∈   ||P1 (ξ1 ) − P2 (ξ2 )|| Lq (Xγ

, g)  const M ξ1 | Bd () − ξ2 | Bd ()  BL p + const M · 2−d (||ξ1 || BL p + ||ξ2 || BL p ), where const M are positive constants depending only on M. (Especially they are independent of  and the integer d > 0.) In particular, if ξ1 | X Bd () = ξ2 | X Bd () then ||P1 (ξ1 ) − P2 (ξ2 )|| Lq (Xγ

, g)  const M · 2−d (||ξ1 || BL p + ||ξ2 || BL p ). Proof Pi (ξi ) = Qi (1 + Ri )−1 ξi ,   = Qi 1 − Ri + Ri2 − · · · + (−1)d−1 Rid−1 ξi + (−1)d Qi Rid (1 + Ri )−1 ξi . From the definitions of the operators Q and R in Section 4, we have Qi Rik ξi = Qi Rik (ξi | X Bd () ) and Q2 Rk2 (ξ2 | X Bd () ) = Q1 Rk1 (ξ2 | X Bd () ) over X for k  d − 1. (These follows from the fact that Qi and Ri have ‘one-step propagation’.) Therefore for γ ∈    ||P1 (ξ1 ) − P2 (ξ2 )|| Lq (Xγ

, g)  const M ξ1 | Bd () − ξ2 | Bd ()  BL p + const M · 2−d (||ξ1 || BL p + ||ξ2 || BL p ). Here we have used ||Ri ||  1/2 (see Lemma 4.1).

 

The following will be used in Section 7. Corollary 6.8 For any ε > 0, there exists d = d(M, ε) > 0 satisfying the following: Let  ⊂  be a finite subset. If E1γ = E2γ , A1γ = A2γ for all γ ∈ Bd ()

Gauge Theory and Mean Dimension

361

and ρ1γ ,s = ρ2γ ,s for all γ ∈ Bd () and s ∈ S with γ s ∈ Bd (), then for any γ ∈ || A(θ1 ) − A(θ2 )|| Lq (Xγ

,g) < ε. Proof ξi = ξ (θi ) satisfies (i = 1, 2)

  ξi + (Pi (ξi ) ∧ Pi (ξi ))+ = −F + Ai .

Let m and d0 be (large) positive integers which will be fixed later. Set d := md0 and suppose that θ1 = θ2 over Bd (). Since we have A 1 = A 2 over Bd (), we have ξ1 − ξ2 = (P2 (ξ2 ) ∧ P2 (ξ2 ))+ − (P1 (ξ1 ) ∧ P1 (ξ1 ))+ , = ((P2 (ξ2 ) − P1 (ξ1 )) ∧ P2 (ξ2 ))+ + (P1 (ξ1 ) ∧ (P2 (ξ2 ) − P1 (ξ1 )))+ , (59) over Bd (). For k = 1, 2, · · · , m, we set ak :=

sup

γ ∈Bkd0 ()

||ξ1 − ξ2 || L p (Xγ

,g) .

From Remark 4.4, Lemma 6.7 and (59), we have ak  const M · b 2

sup

γ ∈Bkd0 ()

||P2 (ξ2 )− P1 (ξ1 )|| Lq (Xγ

,g)  const M ·b 2 (ak+1 +b 4/ p 2−d0 ),

where const M is independent of k and . Since b > 0 is sufficiently small, we have ak  2−1 ak+1 + 2−d0 . Hence a1  2−m+1 am + 2−d0 +1 . We have am  ||ξ1 || BL p + ||ξ2 || BL p  const M b 4/ p  1. Hence a1  2−m+1 + 2−d0 +1 . We have A1 − A2 = P1 (ξ1 ) − P2 (ξ2 ) over Bd (), and for any γ ∈  (using Lemma 6.7) ||P1 (ξ1 ) − P2 (ξ2 )|| Lq (Xγ

,g)  const M (a1 + b 4/ p · 2−d0 ). We choose m and d0 sufficiently large. Then for γ ∈  ||P1 (ξ1 ) − P2 (ξ2 )|| Lq (Xγ

,g) < ε.   Let [θn ] = [(Enγ , Anγ , ρnγ ,s )γ ∈,s∈S ] ∈ GlD (n = 1, 2, 3, · · · ) be a sequence of equivalence classes of gluing data. Since M is compact, if we take a subsequence, this sequence (pointwisely) converges to a gluing data [θ] = [(Eγ , Aγ , ργ ,s )γ ∈,s∈S ] in the following sense (cf. Section 7). For each γ ∈  there exists n0 (γ ) > 0 and a sequence of gauge transformations gnγ : Enγ →

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Eγ (n  n0 (γ )) such that gnγ (Anγ ) converges to Aγ (in the C ∞ -topology) and −1 gnγ s ρnγ ,s gnγ converges to ργ ,s as n → ∞. For each γ ∈  we can assume that, for n  n0 (γ ), Enγ = Eγ , Anγ = Aγ + α˜ nγ and ρnγ ,s = ργ ,s evnγ ,s (s ∈ S) where αnγ ∈ H 1Aγ and vnγ ,s ∈ (adE)xγ ,s . (See   Sections 3.4.2 and 5.1). Moreover we have limn→∞ αnγ  Lq = 0 and limn→∞ |vnγ ,s | = 0. Therefore the bundle map hnγ : Enγ = Eγ → Eγ given in (34) can be defined for n  n1 (γ ). Here n1 (γ ) is an appropriate large number with n1 (γ )  n0 (γ ), n0 (γ s−1 ) (s ∈ S). For each n  1 there exist a (possibly empty) finite subset n ⊂  (each γ ∈ n satisfies n  n1 (γ )) such that we can define a bundle map hn : En | Xn → E| Xn'by gluing these hnγ . We can take these n so that 1 ⊂ 2 ⊂ 3 ⊂ · · · and n1 n = . Let ξn ∈ BL p (+ (adEn )) (n = 1, 2, · · · ) and suppose supn ||ξn || BL p < ∞. For each finite subset  ⊂ , L p (X ) is a reflexive Banach space. Hence if we take a subsequence of {ξn }, there exists ξ ∈ BL p (+ (adE)) with ||ξ || BL p  supn ||ξn || BL p such that, for any finite subset  ⊂ , hn (ξn )| weakly converges to ξ | in L p (X ). Lemma 6.9 In the above situation, hn (Pθn (ξn ))| weakly converges to Pθ (ξ )| in Lq (X ) as n → ∞ for any finite subset  ⊂ .

Proof Take ε > 0 and η ∈ (Lq (X ))∗ = Lq (X ) (1/q + 1/q = 1). Let d > 0 be a large integer which will be fixed later. Set ξn := ξn | Bd () = 1 X Bd () · ξn and ξn

:= ξn − ξn where 1 X Bd () is the characteristic function of X Bd () . We also define ξ := ξ | Bd () and ξ

:= ξ − ξ . hn (ξn ) weakly converges to ξ in L p (X Bd () ). Then P(hn (ξn )) weakly converges to P(ξ ) in BLq . Set Pn := p + q hn ◦ Pn ◦ h−1 n : L (X Bd () ,  (adE)) → BL . (Pn := Pθn .) We have           hn (Pn (ξn )) = Pn hn ξn − P hn ξn + P hn ξn + hn Pn ξn

.

(60)

  hn (Pn (ξ

)) q  const,M · 2−d ||ξn || BL p and From Lemma 6.7, n L (X )    −d −d  P(ξ

) q  const,M · 2 ||ξ || BL p  const,M · 2 supn ||ξn || BL p . L (X ) The term (Pn (hn ξn ) − P(hn ξn )) can be evaluated by using Lemmas 5.1 and 6.7 as follows. Define (for n  0) a gluing data θˆn = ( Eˆ nγ , Aˆ nγ , ρˆnγ ,s )γ ∈,s∈S by ( Eˆ nγ , Aˆ nγ , ρˆnγ ,s ) = (Enγ , Anγ , ρnγ ,s ) for (γ , s) ∈ Bd+1 () × S and ( Eˆ nγ , Aˆ nγ , ρˆnγ ,s ) = (Eγ , Aγ , ργ ,s ) otherwise. Lemma 6.7 gives ( Pˆ n := hn ◦ Pθˆn ◦ h−1 n )       Pn hn ξn − Pˆ n hn ξn 

Lq (X ,g)

 const,M · 2−d ||ξn || BL p .

Lemma 5.1 gives (n  1)      ˆ   Pn hn ξn − P(hn ξn )

BLq

 const M ·

sup

{d Lq ([Anγ ], [Aγ ])

γ ∈Bd+1 (),s∈S

+ |ρnγ ,s − ργ ,s |} ||ξn || BL p

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Therefore for η ∈ (Lq (X ))∗ |hn (Pn (ξn )) − P(ξ ), η|  |P(hn ξn ) − P(ξ ), η| + const,M · ||η|| 2−d sup ||ξm || BL p , m

+ const,M · ||η||

sup

{d Lq ([Anγ ], [Aγ ])

γ ∈Bd+1 (),s∈S

+ |ρnγ ,s − ργ ,s |} sup ||ξm || BL p . m

We choose d > 0 so that const,M · ||η|| 2 n1 > 0 so that for n  n1 const,M ·||η|| P(hn ξn )

Since n  n2

sup

−d

supn ||ξn || BL p  ε/3. We can choose

{d Lq ([Anγ ], [Aγ ])+|ρnγ ,s −ργ ,s |}·sup ||ξm || BL p  ε/3.

γ ∈Bd+1 (),s∈S

m

weakly converges to P(ξ ) in BL , we can choose n2 so that for q

|P(hn ξn )− P(ξ ), η|  ε/3. Therefore for n  max(n1 , n2 ) |hn (Pn (ξn ))− P(ξ ), η|  ε. Thus limn→∞ hn (Pn (ξn )), η = P(ξ ), η. This means that hn (Pn (ξn ))| weakly converges to P(ξ )| in Lq (X ).   6.3 Proof of Surjectivity Let (E1 , A1 ) and (E2 , A2 ) be two pairs of a principal SU(2)-bundle over X and an ASD connection on it. We define the Lq -distance between their gauge equivalence classes by (recall q > 4) d Lq ([E1 , A1 ], [E2 , A2 ]) :=

inf

g:E1 →E2

||A2 − g(A1 )|| Lq (X) ,

where g runs over bundle isomorphisms between E1 and E2 . If E1 and E2 are not isomorphic, then we set d Lq ([E1 , A1 ], [E2 , A2 ]) := ∞. Recall that M denotes a set of gauge equivalence classes of (E, A) satisfying the conditions (a), (b), (c) in the beginning of Section 3.2. Let L ⊂ M be a subset such that there exists δ > 0 satisfying Bδ (L) ⊂ M. Here Bδ (L) ⊂ M means that if a pair (E, A) of a principal SU(2)-bundle E over X and an ASD connection A on E satisfies d Lq ([E, A], [F, B])  δ for some [F, B] ∈ L then [E, A] ∈ M. We define GlD(L) ⊂ GlD by (! " )  GlD(L) := Eγ , Aγ , ργ ,s γ ∈,s∈S ∈ GlD| [Eγ , Aγ ] ∈ L for all γ ∈  Let B be the set of all gauge equivalence classes of (F, B) where F is a principal SU(2)-bundle over X (,S) and B is a connection on it. By using the cut-off construction in Section 3.3, we have the map: J : GlD → B ,

[θ] → [E(θ ), A (θ )].

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For [Fi , Bi ] ∈ B (i = 1, 2), we define their BLq -distance by   d BLq [F1 , B1 ], [F2 , B2 ] := inf ||B2 − g(B1 )|| BLq . g:F1 →F2

(This may be +∞.) For ν > 0 we define a subset U(L, ν) ⊂ B by       U(L, ν) := [F, B] ∈ B | d BLq [F, B], J(GlD(L)) < ν,  F B+  BL p < ν 3/2 . Here d BLq ([F, B], J(GlD(L))) < ν means that there exists a gluing data [θ] ∈ GlD(L) such that d BLq ([F, B], [E(θ), A (θ)]) < ν. The following lemma will be used in Section 8. (This is essentially given in Donaldson and Kronheimer [5, Lemma (7.2.43)].) Lemma √ 6.10 There exists b0 = b0 (M, ν) > 0 and ν = ν (ν) > 0 such that if b = 4N λ  b0 and [F, B] ∈ B satisfies for all γ ∈    +     F  p < ν 3/2 , inf d Lq F| Xγ

, B| Xγ

, E| Xγ

, A| Xγ

< ν , B BL [E,A]∈L

then we have [F, B] ∈ U(L, ν). Proof There are [Eγ , Aγ ] ∈ M (γ ∈ ) and bundle maps gγ : F| Xγ

→ Eγ | Xγ

such that   gγ (B) − Aγ  q

< ν . L (X ,g) γ

From (14) we get

   Aγ − A  q

 const M · b 1+4/q . γ L (X ,g) γ

Hence

  gγ (B) − A  q

< ν + const M · b 1+4/q . γ L (X ,g) γ

For each γ ∈  and s ∈ S, we set hγ ,s := gγ s gγ−1 : Eγ → Eγ s over the ‘neck’ Wγ ,s :=     Xγ

∩ Xγ

s . Then hγ ,s (A γ )− A γ s  Lq (W ,g)  2 ν +const M · b 1+4/q =: ε. In the γ ,s

exponential gauges of Aγ around xγ ,s and yγ ,s , the connection matrix A γ = 0   over the necks. Therefore, in these gauges, dhγ ,s  Lq (Wγ ,s ,g)  ε. Using the q

Sobolev embedding L1 → C0,1−4/q , we get   hγ ,s (x) − hγ ,s (y)  const · ε|x − y|1−4/q ,

for any x, y ∈ Wγ ,s . (The above ‘const’ does not depend on λ.) Since the right-hand-side is sufficiently small, there is ργ ,s : (Eγ )xγ ,s → (Eγ s ) yγ s,s such that hγ ,s = ργ ,s euγ ,s and uγ ,s satisfies   duγ ,s  q  const · ε, |uγ ,s |  const · λ1/2−2/q ε. (61) L (Wγ ,s ,g) Set θ := (Eγ , Aγ , ργ ,s )γ ,s .

Gauge Theory and Mean Dimension

365

We define kγ : Eγ | Xγ

→ Eγ | Xγ

as follows; kγ is equal to e(1−βγ )uγ ,s around  

the points xγ ,s , and equal to e−(1−βγ )u˜ γ ,s u˜ γ ,s = ργ s−1 ,s uγ s−1 ,s ργ−1s−1 ,s around the points yγ ,s . kγ is equal to 1 outside the ‘neck’ regions. We set g˜ γ := kγ gγ : F| Xγ

→ Eγ | Xγ

. These compatibly (i.e. g˜ γ s = ργ ,s g˜ γ ) define g˜ : F → E(θ ). We have g˜ γ (B) − A γ = kγ (gγ (B) − A γ ) + kγ (A γ ) − A γ . From (61) we have  

−2/q kγ (A )− A  q ε and γ γ L (Wγ ,s ,g)  const · ε. (Note that |dβγ ⊗ uγ ,s |  const · λ    const · ε.) Therefore we have hence dβ ⊗ uγ ,s  q γ

L (Wγ ,s ,g)

  g(B) ˜ − A (θ) BLq  const · ε.   Recall that L ⊂ M satisfies Bδ (L) ⊂ M where Bδ (L) is the δ-neighborhood of L with respect to the Lq -distance. Theorem 6.11 There are ν0 (δ) > 0, N0 > 0 and λ0 (N, ν, δ) > 0 satisfying the following: If ν  ν0 (δ), N  N0 and λ  λ0 (N, ν, δ) then for any [F, B] ∈ U(L, ν) there exist [θ] ∈ GlD and ξ ∈ BL p (+ (E(θ ))) satisfying

 [F, B] = E, A (θ) + Pθ (ξ ) ,

||ξ || BL p  3ν 3/2 .

In particular if [F, B] ∈ U(L, ν) and B is an ASD connection, then there exists [θ] ∈ GlD satisfying [F, B] = [E(θ), A(θ)] (see Proposition 4.3 and the statement of the uniqueness of ξ there). We will prove this theorem by using the continuity method developed in Donaldson and Kronheimer [5, Section 7.2.4, 7.2.5].

L)) satisfying B = B + b Let [F,  B] ∈ U(L, ν). There is [F, B ] ∈ J(GlD(

    with b BLq < ν. For t ∈ [0, 1] we set Bt := B + tb. For small ν > 0 and √ b = 4N  + λ > 0, all Bt are contained in U(L, ν); In fact, when t = 0,  F (B0 ) p =  F + (B ) p  const M · b 4/ p < ν 3/2 . For t ∈ (0, 1], F + (Bt ) = BL BL tF + (B) + (1 − t)F + (B ) + (t2 − t)(b ∧ b)+ .         +  F (Bt ) p  t  F + (B) p + (1 − t)  F + (B ) p + const · (t − t2 )  b2 q , BL BL BL BL  4/ p  3/2 2 3/2 < t · ν + const M · (1 − t) b +ν ν . Hence [F, Bt ] ∈ U(L, ν). Let ε > 0 be a small number which will be fixed later. Let S ⊂ [0, 1] be the set of t ∈ [0, 1] such that there exist a gluing data θt , ξt ∈ BL p (+ (E(θt ))) and a gauge transformation ut : F → E(θt ) satisfying ut (Bt ) = A (θt ) + Pθt (ξt ),

||ξt || BL p < ε.

(62)

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M. Tsukamoto

We have 0 ∈ S. From this equation, we have ut (F + (Bt )) = F + ( A (θt )) + ξt + (Pθt ξt ∧ Pθt ξt )+ . Hence     ||ξt || BL p   F + (Bt ) BL p +  F + ( A (θt )) BL p + const M · ||ξt ||2BL p ,  ν 3/2 + const M · b 4/ p + const M · ε ||ξt || BL p .

We choose ε > 0 so that const M · ε  1/2. Then   ||ξt || BL p  2 ν 3/2 + const M b 4/ p . We choose ν and b sufficiently small so that 2(ν 3/2 + const M b 4/ p )  3ν 3/2  ε/2. Therefore we get ||ξt || BL p  3ν 3/2  ε/2.

(63)

In particular, from the open condition ||ξt || BL p < ε, we have deduced the closed condition ||ξt || BL p  ε/2. We will prove that S is a closed set in [0, 1]. Let tn ∈ S (n = 1, 2, 3, · · · ) be a sequence converging to t ∈ [0, 1]. Set θn = θtn = (Enγ , Anγ , ρnγ ,s )γ ∈,s∈S . We have un (Btn ) = A n + Pn (ξn ) with ||ξn || BL p  ε/2. From the argument before Lemma 6.9, using some gauge transformations, we can suppose that θn converges to θ = (Eγ , Aγ , ργ ,s )γ ∈,s∈S as follows; There is n0 (γ ) > 0 for each γ ∈  such that Enγ = Eγ , Anγ = Aγ + α˜ nγ (αnγ ∈ H 1Aγ ), ρnγ ,s = ργ ,s evnγ ,s for n  n0 (γ ), and αnγ and vnγ ,s converge to 0 as n → ∞. Moreover there exist ξ ∈ BL p (+ (adE)), an exhausting sequence 1 ⊂ 2 ⊂ 3 ⊂ · · · ⊂  and bundle maps hn : En | Xn → E| Xn such that hn (ξn )| X weakly converges to ξ | X in L p (X ) for any finite subset  ⊂ . From ||ξn || BL p  ε/2, we have ||ξ || BL p  ε/2. Set gn := hn ◦ un (over Xn ). Then     over Xn . gn (Bn ) = hn A n + hn Pn (ξn ) For any finite subset  ⊂ , the right-hand-side of this equation weakly converges to A (θ) + Pθ (ξ ) in Lq (X ) (Lemma 6.9). On the other hand, if we take a subsequence, there exists a bundle map g defined over X (,S) such that q gn weakly converges to g in L1 (X ) for each finite subset  ⊂ . Then we get g(Bt ) = A (θ) + Pθ (ξ ),

||ξ || BL p  ε/2 < ε.

This shows t ∈ S. Thus S is a closed set in [0, 1]. Next we will prove that S is open in [0, 1]. Suppose that the equation (62) holds at some t ∈ [0, 1]. Then A = A (θt ) satisfies d BLq ([ A ], [B ])  ν + const · ε. Therefore if we choose b , ε and ν small enough, then θt = (Eγ , Aγ , ργ ,s )γ ∈,s∈S satisfies Bδ/2 ([Eγ , Aγ ]) ⊂ M, for every γ ∈ . (Recall that Bδ (L) ⊂ M.)

(64)

Gauge Theory and Mean Dimension

367

Consider the following map: G : B1 → B2 ,

  (χ, v, α, η) → e−χ A v,α + Pv,α (η + ξt ) − ut (Bt ),

where B1 and B2 denote the Banach spaces defined in Section 6.1. Of course, we consider only very small (χ, v, α, η) ∈ B1 . A v,α and Pv,α are the connection and the operator defined in Section 5.1. A 0,0 = A = A (θt ) and P0,0 = Pθt . We have G(0) = 0. If we prove that the derivative of G at the origin (dG)0 : B1 → B2 is isomorphic, then the inverse mapping theorem and (64) implies that t ∈ S is an inner point. (dG)0 : B1 → B2 is given by (dG)0 (χ, v, α, η) = T(χ, v, α, η) + [P(ξt ), χ] + ∂v Pv,0 (ξt ) + ∂α P0,α (ξt ), where T(χ, v, α, η) = d A χ + j1 (v) + j2 (α) + P(η) is the operator given in (48) and P = P0,0 = Pθt . Proposition 6.6 shows that T is an isomorphism satisfying ||(χ, v, α, η)|| B1  K ||T(χ, v, α, η)|| B2 . Therefore if we prove ||T − (dG)0 || < K−1 , then (dG)0 is an isomorphism. We have ||[P(ξt ), χ]|| BLq  const M ·||ξt || BL p ||χ||C 0  const M ·ε ||(χ, v, α, η)|| B1 from Lemma 6.2 and (46). We have d+A [P(ξt ), χ] = [ξt , χ] − [P(ξt ) ∧ d A χ]+ . Hence  +  d [P(ξt ), χ] p  ||χ||C 0 ||ξt || BL p + ||P(ξt )|| BLq ||d A χ|| BL4 . A BL We have ||d A χ|| BL4  ||d A χ + j1 (v)+ j2 (α)|| BL4 +|| j1 (v)|| BL4 +|| j2 (α)|| BL  4 . Recall  the equation (45) ( j1 (v) = dβγ ⊗ vγ ,s over (xγ ,s )) and the fact that dβγ  L4 is a constant independent of the parameters λ and N (because of the conformal invariance of the L4 -norms of 1-forms; see the argument before (17)). Hence (using Lemma 6.2) || j1 (v)|| BL4  const ||v||  const M ||(χ, v, α, η)|| B1 .

(65)

From Lemma 6.1, || j2 (α)||  BL4  const  · || j2 (α)|| BLq  const M · ||α||  const M · ||(χ, v, α, η)|| B1 . Hence d+A [P(ξt ), χ] BL p  const M · ε ||(χ, v, α, η)|| B1 . Thus ||[P(ξt ), χ]|| B2  const M · ε ||(χ, v, α, η)|| B1 . By using an argument similar to that in Lemma 5.1, we have     ∂v Pv,0 (ξt ) q + ∂α P0,α (ξt ) q  const M · ε(||v|| + ||α||) BL BL  const M · ε ||(χ, v, α, η)|| B1 .

Differentiating the equation d+A v,α Pv,α (ξt ) = ξt with respect to v at (v, α) = (0, 0),  + we have d+A ∂v Pv,0 (ξt ) = − j1 (v) ∧ P(ξt ) . Using the above (65), we have  +  d ∂v Pv,0 (ξt ) p  || j1 (v)|| 4 ||P(ξt )|| BLq  const M · ε ||(χ, v, α, η)|| B . BL A 1 BL Similarly   + d ∂α P0,α (ξt ) p  const M · ε ||α||  const M · ε ||(χ, v, α, η)|| B . A 1 BL

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M. Tsukamoto

Therefore     ∂v Pv,0 (ξt ) + ∂α P0,α (ξt )  const M · ε ||(χ, v, α, η)|| B . 1 B2 B2 Thus ||(dG)0 (χ, v, α, η) − T(χ, v, α, η)|| B2  const M · ε ||(χ, v, α, η)|| B1 . Hence if we choose ε > 0 small enough, then (dG)0 is an isomorphism. This shows that S is open in [0, 1]. S is a non-empty open closed set in [0, 1]. Thus S = [0, 1]. In particular we have 1 ∈ S. This proves Theorem 6.11. 7 Estimation of the Mean Dimension As in the previous sections, M denotes a set of equivalence classes of (E, A) (E is a principal SU(2)-bundle over X and A is an ASD connection on it) which satisfies the conditions (a), (b), (c) in the beginning of Section 3.2. GlD = GlD M is the set of the equivalence classes of M-gluing data defined in Definition 3.2. GlD is endowed with the topology of ‘point-wise convergence’ as follows. A sequence [θn ] = [Enγ , Anγ , ρnγ ,s ]γ ∈,s∈S in GlD (n  1) converges to [θ] = [Eγ , Aγ , ργ ,s ]γ ∈,s∈S if the following condition is satisfied. For any finite subset  ⊂ , there exist n0 () > 0 and bundle isomorphisms gnγ : Enγ → Eγ for n  n0 () and γ ∈  such that gnγ (Anγ ) converges to Aγ (as n → ∞) −1 in the C ∞ -topology for all γ ∈ , and that gnγ s ρnγ ,s gnγ : (Eγ )xγ ,s → (Eγ s ) yγ s,s converges to ργ ,s for any (γ , s) with γ , γ s ∈ . This topology is metrizable and compact (because we suppose that M is compact).  continuously acts on GlD by (this is a right action)

  Eγ , Aγ , ργ ,s γ ∈,s∈S · g := Egγ , Agγ , ρgγ ,s γ ∈,s∈S , where we naturally consider that (Egγ , Agγ ) is a data defined on Xγ and that ρgγ ,s is a map from (Egγ )xγ ,s to (Egγ ) yγ s,s . A distance on GlD is given as follows: For n  1, [θ] = [Eγ , Aγ , ργ ,s ]γ ∈,s∈S and [θ ] = [Fγ , Bγ , ργ ,s ]γ ∈,s∈S in GlD, we define δn ([θ], [θ ]) by ⎛   gγ (Aγ ) − Bγ  ∞ ⎝ δn ([θ], [θ ]) := inf L (X) gγ :Eγ →Fγ (γ ∈Bn )

γ ∈Bn

+



⎞ |gγ s ργ ,s gγ−1 − ργ ,s |⎠ ,

γ ,γ s∈Bn

where Bn is the n-ball (with respect to the word distance) centered at the origin in  and gγ (γ ∈ Bn ) runs over bundle isomorphisms between Eγ and Fγ . If Eγ is not isomorphic to Fγ for some γ ∈ Bn , then we set δn ([θ ], [θ ]) := +∞. We define a distance d([θ], [θ ]) by 1 δn ([θ], [θ ]) d([θ], [θ ]) := . 2n 1 + δn ([θ ], [θ ]) n1

Gauge Theory and Mean Dimension

369

We define the space M(GlD) = M(GlD M ) by   M(GlD) := [E(θ), A(θ)]| [θ ] ∈ GlD . M(GlD) is endowed with the topology of C ∞ -convergence on compact subsets in X (,S) . This topology is metrizable.  continuously acts on M(GlD) by (3). The map

GlD → M(GlD),

[θ] → [E(θ ), A(θ )],

is -equivariant. Lemma 7.1 The above map GlD → M(GlD) is a -homeomorphism. Proof Proposition 5.5 shows that the map is bijective. Since GlD is compact, it is enough to prove that the map is continuous.   Let ε > 0 and  ⊂  be a finite subset. Let θ1 = E1γ , A1γ , ρ1γ ,s γ ∈,s∈S and   θ2 = E2γ , A2γ , ρ2γ ,s γ ∈,s∈S be two gluing data. Let d = d(M, ε) be the positive constant given by Corollary 6.8. Suppose that E1γ = E2γ for γ ∈ Bd+1 () and    that  A1γ − A2γ  Lq and |ρ1γ ,s − ρ2γ ,s | γ ∈ Bd (), s ∈ S are sufficiently small.     We define another gluing data θ := Eγ , A γ , ργ ,s γ ∈,s∈S by Eγ , A γ , ργ ,s =     (E1γ , A1γ , ρ1γ ,s ) for γ ∈ Bd () and Eγ , A γ , ργ ,s = E2γ , A2γ , ρ2γ ,s for γ ∈  \ Bd (). From Corollary 6.8, we have (for γ ∈ )    A(θ1 ) − A(θ ) q

< ε. L (X ,g) γ

On the other hand, for all γ ∈  and s ∈ S we have E2γ = Eγ , A2γ − A γ Lq (X

,g)  1 and |ρ2γ ,s − ργ ,s |  1. Therefore (using the arguments in γ

Section 5.1)

  d BLq [ A(θ2 )], [ A(θ )] < ε.

Thus there exists a bundle map g from E(θ1 ) to E(θ2 ) over X such that for all γ ∈  ||g( A(θ1 )) − A(θ2 )|| Lq (Xγ

,g) < 2ε. This shows that GlD → M(GlD) is continuous.

 

In the rest of this section we assume that  is amenable. Let 1 ⊂ 2 ⊂ 3 ⊂ · · · be an amenable sequence in . This sequence satisfies (for any r > 0)    Br (n )|/|n  → 1 (n → ∞), (66) where Br (n ) is the r-neighborhood of n . For each n we define the distance dn ([θ], [θ ]) on GlD by     dn [θ], [θ ] := max d [θ].g, [θ ].g . g∈n

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M. Tsukamoto

Proposition 7.2

  dim M(GlD) :   3|S| + dim M,

where dim M denotes the (topological) covering dimension of M. Proof From Lemma 7.1, it is enough to prove that dim(GlD : )  3|S| + dim M. Fix any ε > 0. Take n0 = n0 (ε) > 0 satisfying 1 < ε. 2n nn 0

For any finite set  ⊂ , we define B−1 () as the set of γ ∈  satisfying γ s ∈  for all s ∈ S. We define a finite dimensional compact metrizable space GlD| by (  GlD| := (Eγ , Aγ )γ ∈ , (ργ ,s )γ ∈B−1 (),s∈S | [Eγ , Aγ ] ∈ M, ) ργ ,s : (Eγ )xs → (Eγ s ) ys / ∼,  where the equivalence relation ∼ is defined as follows. θ = (Eγ , Aγ )γ ∈ ,    (ργ ,s )γ ∈B−1 (),s∈S is equivalent to θ = (Fγ , Bγ )γ ∈ , (ργ ,s )γ ∈B−1 (),s∈S if there exist gγ : Eγ → Fγ (γ ∈ ) such that gγ (Aγ ) = Bγ for all γ ∈  and ργ ,s gγ = gγ s ργ ,s for all (γ , s) ∈ B−1 () × S. There is a natural projection GlD → GlD| . Consider the following projection map: GlD| → M . The topological dimension of each fiber of this map is  3|||S|. Hence dim GlD|  || dim M + 3|||S|. For each n in the amenable sequence, consider the following map p : GlD → GlD| Bn0 (n ) .

If p([θ]) = p([θ ]), then we have dn ([θ], [θ ]) < ε. Therefore (see Appendix B)

  Widimε (GlD, dn )  dim GlD| Bn0 (n )  |Bn0 (n )|(dim M + 3|S|).

Using (66), we get Widimε (GlD : ) = lim Widimε (GlD, dn )/|n |  dim M + 3|S|. n→∞

This holds for any ε > 0. Thus we get the above result.

 

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Let [E, A] ∈ M. We call this point a regular point of M if [E, A] ∈ M0 and there is δ > 0 such that, for any ASD connection B on E satisfying d Lq ([A], [B])  δ, the pair [E, B] is contained in M0 . Remember that for any [E, A] ∈ M0 the connection A is irreducible (see (b) in the beginning of Section 3.2) and then its isotropy group is {±1}. Proposition 7.3 Let [E, A] be a regular point of M, then we have dim(M(GlD) : )  3|S| + dim H 1A . Proof We will prove dim(GlD : )  3|S| + dim H 1A . There is a compact ˜ ∈ neighborhood K of the origin in H 1A such that, for all α ∈ K, [E, A + α] M0 and the map K α → [E, A + α] ˜ ∈ M0 is injective. Here A + α˜ is the ASD connection introduced in Section 3.4.2. Let Hom SU(2) (Exs , E ys ) be the space of SU(2)-isomorphisms between the fibers Exs and E ys (s ∈ S). Let Ls ⊂ Hom SU(2) (Exs , E ys ) be a compact set such that Ls is homeomorphic to a three dimensional ball and that, for any u, v ∈ Ls , we have u = −v. Then we have a natural -equivariant continuous map   # K× Ls → GlD. s∈S

From the conditions of K and Ls , this map is injective. Therefore ⎛ ⎞    # # ⎝ ⎠ K× Ls :  = dim K × Ls dim(GlD : )  dim s∈S

s∈S

= dim H 1A + 3|S|.  

8 Proof of Theorem 2.3 In this and the next sections we set X := S4 with the metric h satisfying the conditions (i), (ii) in the beginning of Section 2.1. Let 0  c < c < +∞ and d ∈ (2, +∞]. As in Section 2.1 we define M = MS4 (c, d ) as the space of the gauge equivalence classes of (E, A) where E is a principal SU(2)-bundle over S4 and A is an ASD connection on it satisfying ||F A || Ld (S4 ,h)  c. We set M1 := {[S4 × SU(2), the product connection]} and M0 := M \ M1 . If A is an ASD connection on (S4 , h), then A is also ASD with respect to the standard metric on S4 because h is conformally equivalent to the standard one. Therefore all non-flat ASD connections on (S4 , h) are regular. In particular, M = M0  M1 satisfies the conditions (a), (b), (c) in the beginning of Section 3.2. We consider the gluing data space GlD = GlD M for this M = MS4 (c, d ).

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Recall that M(c, d ) is the space of [E, A] where E is a principal SU(2)bundle over (S4 )(,S) and A is an ASD connection on it satisfying (2): ||F A || Ld (Xγ

,g)  c

for all γ ∈ .

(Here Xγ

= U γ in the notation of Section 2.1.) Proposition 8.1 There are N0 (c, c, d ) > 0 and λ0 (c, c, d, N) > 0 such that if N  N0 (c, c, d ) and λ  λ0 (c, c, d, N) then M(c, d ) ⊂ M(GlD),

i.e., for any [E, A] ∈ M(c, d ) there exists [θ ] ∈ GlD satisfying [E, A] = [E(θ), A(θ)]. We need some preliminary results for the proof of this proposition. Our argument is based on Donaldson and Kronheimer [5, Section 7.3]. The following proposition is given in [5, Proposition 7.3.3], and we omit the proof. Proposition 8.2 Let T > 0 and k > 0. Consider (−T, T) × S3 with the usual product metric. There are positive constants η and C = C(k) (η is independent of T and k, and C is independent of T) such that if an ASD connection A on (−T, T) × S3 satisfies  2 ||F(A)|| L2 := |F(A)|2 dvol  η2 , (−T,T)×S3

then |F(A)|  Ce2(|t|−T) ||F(A)|| L2 , for all (t, θ) ∈ (−T, T) × S3 with |t|  T − k. Using the stereographic projection, we can translate this proposition to a result on the Euclidean space: √ Corollary 8.3 Let σ > 0 and λ > 0 with λ/σ  λ/2. Set k := 0.9 and    := x ∈ R4 | kλ/σ < |x| < k−1 σ . There exist η > 0 and C > 0 (independent of σ and λ) such that if an ASD connection A on  satisfies  2 ||F(A)|| L2 := |F(A)|2 dvol  η2 , 

then

 √ C ||F(A)|| λ/2  |x|  σ . 2 L σ2 Moreover A can be represented by the connection matrix satisfying  √ C|x| |A|  2 ||F(A)|| L2 λ/2  |x|  σ . σ |F(A)| 

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Proof See Donaldson and Kronheimer [5, p. 314].

Lemma 8.4 For any ν > 0 there is λ0 > 0 such that if λ  λ0 then all [E, B] ∈ M(c, d ) satisfies inf

[A]∈MS4 (c,d )

  d Lq [B| Xγ

], [A| Xγ

] < ν

for all γ ∈ .

Note that [A] runs over MS4 (c, d ) (not M = MS4 (c, d )). Proof We can prove this lemma by using the argument in [5, Section 7.3.4]. For ε > 0 we set (cf. Section 3.4.3) Xε := S4 \

 

 ¯ s , ε) . ¯ s , ε) ∪ B(y B(x

s∈S

Suppose the above statement is false. Then there are ν > 0 and a decreasing sequence λ1 > λ2 > λ3 > · · · → 0 and ASD connections Bn on Xλn /σ (σ is a small positive constant chosen below) satisfying ||F(Bn )|| Ld (X√λn /2 ,g)  c,

inf

[A]∈MS4 (c,d )

(67)

   d Lq Bn | X√λn /2 , A| X√λn /2  ν.

(68)

Let  n (xs ) and  n (ys ) (s ∈ S) be the annulus regions (in X) around xs and ys of inner radius = kλn /σ and outer radius = σ . Since d > 2, we can choose σ > 0 so small that all Bn and [A] ∈ MS4 (c, d ) have curvatures of L2 -norm  η over each  n (xs ) and  n (ys ). (η is a positive constant given in Corollary 8.3.) From (67) and d > 2, the Uhlenbeck compactness implies that (if we choose a subsequence) there exists [A] ∈ MS4 (c, d ) such that [Bn ] converges to [A] in the C ∞ -topology over compact subsets in X \ {xs , ys | s ∈ S}. On the other hand, from Corollary 8.3, Bn and A can be represented over  n (xs ) and  n (ys ) by the connection matrices satisfying |Bn |  const · |x|,

|A|  const · |x|

*  λn /2  |x|  σ ,

where ‘const’ is a positive constant independent of λn . This estimate and the C ∞ -convergence mentioned above imply    d Lq Bn | X√λn /2 , A| X√λn /2 → 0. This contradicts (68).

 

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Proof of Proposition 8.1 Set L := MS4 (c, d )  M = MS4 (c, d ). For any ν > 0, Lemmas 6.10 and 8.4 imply (for appropriate N and λ) M(c, d ) ⊂ U(L, ν). Then, from Theorem 6.11, for any [E, A] ∈ M(c, d ) there exists [θ] ∈ GlD M satisfying [E, A] = [E(θ), A(θ)].   Proof of Theorem 2.3 ¯ d ) = M1 i.e., M (i) Take c¯ such that 0  c < c¯ < c0 (d ). Then M := MS4 (c, consists only of the product connection. Then all A(θ ) ([θ ] ∈ GlD M ) become flat connections. (See Remark 4.5.) Since we have M(c, d ) ⊂ M(GlD M ) (for appropriate N and λ), M(c, d ) is equal to the moduli space of flat SU(2)-connections. ¯ d ) (the constant in Proposition 8.1). Using Propositions (ii) Fix N = N0 (c, c, 7.2 and 8.1, we get (λ  1) dim(M(c, d ) : )  dim(M(GlD) : )  3|S| + dim MS4 (c, d ).  

9 Proof of Theorem 2.4 In this section we suppose X = S4 and d ∈ (2, +∞). Let 0 < c < c < +∞ and set c := (c + c)/2 (c < c < c). We also suppose that dim MS4 (c, d ) > 0. Then there exists [E0 , A0 ] ∈ MS4 (c, d ) such that A0 is a regular ASD connection and dim H 1A0  dim MS4 (c, d ). [E0 , A0 ] becomes a regular point of M := MS4 (c , d ). (See Proposition 7.3.) √ Proposition 9.1 There is b0 (c, c , d ) > 0 such that if b = 4N λ  b0 (c, c , d ) then M(GlD M ) ⊂ M(c, d ),

i.e., for any [θ] ∈ GlD M we have ||F( A(θ))|| Ld (Xγ

,g)  c

for all γ ∈ .

Proof We use an argument similar to the proof of Lemma 8.4. Set ε := (c − c )/2 = (c − c)/4. Suppose the above statement is false. Then there are

) is the posiparameters λn > 0 and Nn  N0 (M ) (n = 1, 2, 3, · · · ) (N0 (M√ tive constant given by Proposition 4.3) satisfying b n := 4Nn λn → 0, and M -gluing data θn = (Enγ , Anγ , ρnγ ,s )γ ∈,s∈S such that for some γn ∈    (n)   F A (θn )  d

> c, L (X ,g) γn

(n)

(n)

where A := A (θn ) is A(θn ) for the parameters λ = λn and N = Nn . Using the -equivariance, we can assume that γn = e (the identity element of ). Taking a subsequence, we can also assume that Ene = Eme (=: E) for m  n  1 and Ane converges to A (an ASD connection on E) with [E, A] ∈ M in

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the C∞ -topology. Since bn → 0, A(n) converges to A up to gauge equivalence in the C ∞ -topology over compact subsets  in Xe \ {xe,s , ye,s | s ∈ S} We have a uniform upper-bound on  F( A(n) ) BL p by Proposition 4.6. Then, from p > 2 and Corollary 8.3, there exists σ > 0 such that in the Euclidean √ coordinates around xe,s and ye,s we have |F( A(n) )|(x)  constσ for λn /2 < |x| < σ (for all n). Hence for a sufficiently small σ < σ we have (recall: ε = (c − c )/2)    (n) d  F A   εd . √ λn /2 <|x|<σ

(Here we have used d < +∞.) |F( A(n) )| uniformly converges to |F(A)| over ¯ e,s , σ ) and B(y ¯ e,s , σ ) in Xe,σ . (Xe,σ is the complement of the balls B(x Xe = S4e .) Hence for n  1   (n)   F A  d  c + ε. L (X , g) e,σ

Therefore for n  1

  (n)   F A 

Ld (Xe

, g)

 c + ε + ε = c.

This contradicts the assumption.

 

Proof of Theorem 2.4 Using Propositions 7.3 and 9.1, we get (λ  1) dim(M(c, d ) : )  dim(M(GlD M ) : )  3|S| + dim H 1A0  3|S| + dim MS4 (c, d ).

  Appendix A: Proof of the Compactness of M(c, d ) The purpose of this appendix is to prove the following proposition: Proposition 9.2 Let d > 2. Let X be a connected, oriented (possibly noncompact) Riemannian 4-manifold without boundary, and E be a principal SU(2)-bundle over X. Let {An }n1 be a sequence of ASD connections on E such that for any compact set K ⊂ X we have sup ||F(An )|| Ld (K) < ∞. n1

Then there exist a subsequence (we also denote it by {An }n1 ), a sequence of gauge transformations gn : E → E and an ASD connection A on E such that gn (An ) converges to A in the C ∞ -topology over every compact set in X. This proposition follows from Donaldson and Kronheimer [5, Proposision (4.4.9)]. But, unfortunately, I think that the proof of [5, Proposision (4.4.9)] contains a gap. I think that the proof of [5, Lemma (4.4.5)] is not correct

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(cf. [18, the footnote in p. 5]). The proof given below essentially uses the fact that the gauge group is SU(2), and I don’t know whether the same result holds for more general gauge group. In Section 2, we introduce the moduli space M(c, d ) and state that this space is compact in the topology defined there. This compactness easily follows from the above proposition. If X is a compact manifold, then the above statement is certainly wellknown (see [18, Theorem E]). So we assume that X is non-compact below. Lemma 9.3 Let Y ⊂ X be a compact submanifold with boundary in X satisfying H 4 (X, Y; Z) = 0. Let g be a gauge transformation of E over Y. (Strictly speaking, we suppose that g is defined smoothly over a neighborhood of Y.) Then g can be extended to a gauge transformation of E defined over X. Proof In this proof, we use the fact that the gauge group is SU(2). Since X is non-compact, E is isomorphic to the product bundle X × SU(2). Hence g can be identified with the map from Y to SU(2). SU(2) ∼ = S3 is 2-connected and π3 (SU(2)) = Z. So the obstruction on the extension of g to X is contained in H 4 (X, Y; Z). But this is 0. Hence g can be extended over X.   From the second countability of X, there exists a sequence of compact submanifolds (with boundary) Yk (k  1) such that  X= Yk . Y1  Y2  Y3  · · · , k1

Lemma 9.4 We can choose the above sequence so that H 4 (X, Yk ; Z) = 0 for all k  1. Proof Let X \ int(Yk ) = X1  X2  · · ·  X N be the decomposition into the sum of the connected components. N is less than or equal to the number of the connected components of ∂Yk . Each Xn is a (possibly non-compact) submanifold in X with non-empty boundary (∂Yk = ∂ X1  · · ·  ∂ X N ). We have (by the excision theorem) N   #   H 4 X, Yk ; Z = H 4 Xn , ∂ Xn ; Z . n=1

If Xn is non-compact, then we get H 4 (Xn , ∂ Xn ; Z) = 0. Hence if we can arrange Yk so that all Xn becomes non-compact, then we get H 4 (X, Yk ; Z) = 0. This can be easily achieved as follows: Suppose that X1 , X2 , · · · , Xm are compact and that Xm+1 , Xm+2 , · · · , X N are non-compact. Then we set Yk := Yk ∪ X1 ∪ X2 ∪ · · · ∪ Xm . Yk also becomes a compact submanifold in X, and X \ int(Yk ) = Xm+1 ∪ Xm+2 ∪ · · · ∪ X N . Since each Xn (n  m + 1) is noncompact, we get H 4 (X, Yk ; Z) = 0.   We suppose that the sequence Yk satisfies H 4 (X, Yk ; Z) = 0.

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Proof of Proposition 9.2 Using a collar neighborhood of ∂Yk , we can construct a open set U k ⊃ Yk (in X) which is diffeomorphic to Yk ∪∂Yk ∂Yk × [0, 1). We can arrange them so that U 1 ⊂ U 2 ⊂ U 3 ⊂ · · · . Using [18, Thoerem E’] for these U k and a diagonal process, we get a subsequence (we also denote it by {An }) satisfying the following: For each k  1 there exist a sequence of gauge transformations gn(k) on U k and an ASD connection A(k) defined over U k such that gn(k) (An ) converges to A(k) (n → ∞) in the C ∞ -topology over compact subsets in U k . For each k  1, A(k+1) is gauge equivalent to A(k) over U k . Hence there exists a gauge transformation hk defined over U k satisfying hk (A(k+1) ) = A(k) on U k . Using Lemma 9.3, we have a gauge transformation h k defined over X which is equal to hk over a neighborhood of Yk . We have h k (A(k+1) ) = A(k) over a neighborhood of Yk . We define an ASD connection A over X by setting A := A(1) over Y1 and A := h 1 ◦ h 2 ◦ · · · ◦ h k−1 (A(k) ) over Yk (k  2). This definition is well-defined. For each k  1, by applying Lemma 9.3 to the sequence {h 1 ◦ h 2 ◦ · · · ◦

hk−1 ◦ gn(k) }n1 , we get a sequence of gauge transformations {u(k) n }n1 defined

(k) = h ◦ h ◦ · · · ◦ h ◦ g over Y . Then u(k) over X such that u(k) k n n n (An ) con1 2 k−1 ∞ verges to A (n → ∞) in the C -topology over Yk . In particular, there exists nk  1 satisfying     A − u(k) An  nk

k

C k (Yk )

 1/k.

(Here ||·||C k (Yk ) is a C k -norm over Yk defined by using a fixed connection on E.) We can choose the above nk so that n1 < n2 < n3 < · · · . Then u(k) nk (Ank ) converges to A (k → ∞) in the C ∞ -topology over every compact subset in X.  

Appendix B: Review of Mean Dimension We review the definitions and basic properties of mean dimension. For the detail, see Gromov [8], Lindenstrauss and Weiss [10] and Lindenstrauss [9]. Let (X, d ) be a compact metric space and ε > 0. Let Y be a topological space and f : X → Y a continuous map. We call f an ε-embedding if we have Diam f −1 (y)  ε for all y ∈ Y. For example, consider [0, 1] × [0, ε] with the standard Euclidean distance, and let f : [0, 1] × [0, ε] → [0, 1] be the natural projection. Then f is an ε-embedding. We define Widimε (X, d ) as the minimum integer n  0 such that there exist an n-dimensional polyhedron P and an ε-embedding f : X → P. For example, we have Widimε ([0, 1] × [0, ε], the Euclidean distance) = 1 for ε < 1. The following is one of the most basic examples (see Gromov [8, p. 332]). For its proof, see Lindenstrauss and Weiss [10, Lemma 3.2] or Tsukamoto [15, Example 4.1].

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Example 9.5 Consider [0, 1] N with the ∞ -distance d∞ (x, y) = maxi |xi − yi |. For ε < 1 we have   Widimε [0, 1] N , d∞ = N. Let  be a finitely generated infinite group with a finite generating set S.  is equipped with the word distance: for γ , γ ∈ , we define d(γ , γ ) as the minimum integer n  0 such that there exist γ1 , · · · , γn in S ∪ S−1 satisfying γ −1 γ = γ1 · · · γn . For a finite subset  ⊂  and r > 0, we define the r-boundary ∂r  ⊂  as the set of γ ∈  such that the r-ball B(γ , r) around γ has non-empty intersection with both  and  \ . Let 1 ⊂ 2 ⊂ 3 ⊂ · · · be a sequence of finite subsets in . We call this sequence an amenable sequence if for any r > 0 we have |∂r n |/|n | → 0 as n goes to ∞. We call  an amenable group if it has an amenable sequence. In this appendix we assume that  is an amenable group with an amenable sequence 1 ⊂ 2 ⊂ 3 ⊂ · · · . For example, Z is an amenable group with the amenable sequence n := {0, 1, · · · , n}. Suppose that  continuously (not necessarily isometrically) acts on X. (We suppose that the action is a right-action.) For a finite subset  ⊂  we define the distance d (·, ·) by setting d (x, y) := sup d(x.γ , y.γ ), γ ∈

for x, y ∈ X. For ε > 0 we define Widimε (X : ) by Widimε (X : ) := lim Widimε (X, dn )/|n |. n→∞

This limit always exists, and it is independent of the choice of an amenable sequence. (see Gromov [8, pp. 336–338] and Lindenstrauss and Weiss [10, Appendix]). We define the mean dimension dim(X : ) by dim(X : ) := lim Widimε (X : ). ε→0

(This might be infinity.) The value of dim(X : ) is a topological invariant. That is, if two distances d and d on X define the same topology, then we have dim((X, d ) : ) = dim((X, d ) : ). The following is the most basic result. (See Gromov [8, p. 340] and Lindenstrauss and Weiss [10, Proposition 3.1, 3.3].) Example 9.6 Let K be a compact metric space and set X := K .  acts on X by the shift action: for x = (xγ )γ ∈ ∈ X and g ∈  we set x.g = (yγ )γ ∈ ,

yγ := xgγ .

Then we have dim(X : )  dim K,

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where dim K denotes the topological covering dimension of K. Moreover if K is a finite polyhedron, then we have dim(X : ) = dim K. Proof Set N := dim K and we suppose DiamK = 1 for simplicity. Let w :  → R>0  be a positive function satisfying w(e) = 1 (e is the identity element of ) and γ ∈ w(γ ) = 2. We define the distance d(x, y) (x, y ∈ X) by setting w(γ )d(xγ , yγ ). d(x, y) := γ ∈

For ε > 0, let r > 0 be a positive number such that the sum of w(γ ) over γ ∈  \ B(e, r) is less than ε. Then for any n , the natural projection ϕ : X → Kn ∪∂r n satisfies Diam(ϕ −1 (y), dn ) < ε for any y ∈ Kn ∪∂r n . Therefore Widimε (X, dn )  N|n ∪ ∂r n |. Since limn→∞ |n ∪ ∂r n |/|n | = 1, we have Widimε (X : )  N. Hence dim(X : )  N. Next we suppose K is a polyhedron. Then there exists a topological embedding [0, 1] N → K. So we can assume K = [0, 1] N with the ∞ -distance. There exists a distance non-decreasing continuous map from ([0, 1] N|n | , d∞ ) to (X, dn ). Then for ε < 1     Widimε X, dn  Widimε [0, 1] N|n | , d∞ = N|n |. Here we have used the result of Example 9.5. Hence we get Widimε (X : )  N for ε < 1. Thus dim(X : ) = N.  

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10. Lindenstrauss, E., Weiss, B.: Mean topological dimension. Israel J. Math. 115, 1–24 (2000) 11. Macrì, M., Nolasco, M., Ricciardi, T.: Asymptotics for selfdual vortices on the torus and on the plane: a gluing technique. SIAM J. Math. Anal. 37, 1–16 (2005) 12. Nevanlinna, R.: Zur Theorie der meromorphen Funktionen. Acta Math. 46, 1–99 (1925) 13. Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differential Geom. 17, 139–170 (1982) 14. Tsukamoto, M.: Gluing an infinite number of instantons. Nagoya Math. J. 188, 107–131 (2007) 15. Tsukamoto, M.: Moduli space of Brody curves, energy and mean dimension. Nagoya Math. J. 192, 27–58 (2008) 16. Tsukamoto, M.: Deformation of Brody curves and mean dimension. Ergodic Theory Dynam. Systems. arXiv:0706.2981 (to appear) 17. Uhlenbeck, K.K.: Connections with L p bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982) 18. Wehrheim, K.: Uhlenbeck Compactness, EMS Series of Lectures in Mathematics. European Mathematical Society, Zürich (2004)

Math Phys Anal Geom (2009) 12:381–444 DOI 10.1007/s11040-009-9068-9

Dynamical Localization for Unitary Anderson Models Eman Hamza · Alain Joye · Günter Stolz

Received: 3 March 2009 / Accepted: 9 September 2009 / Published online: 23 September 2009 © Springer Science + Business Media B.V. 2009

Abstract This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of AizenmanMolchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band

The first author was partially supported through MSU New Faculty Grant 07-IRGP-1192, while the last author was partially supported through US-NSF grant DMS-0653374. E. Hamza Department of Physics, Faculty of Science, Cairo University, Cairo 12613, Egypt A. Joye Institut Fourier, Université de Grenoble, BP 74, Saint-Martin d’Hères, 38402, France G. Stolz (B) Department of Mathematics, University of Alabama at Birmingham, CH 452, Birmingham, AL 35294-1170, USA e-mail: [email protected]

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edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes. Keywords Anderson model · Localization · Unitary operators · Fractional moment method Mathematics Subject Classifications (2000) 82B44 · 47B80

1 Introduction The spectral theory of Schrödinger operators and other selfadjoint operators H used to model hamiltonians of quantum mechanical systems has a long history. It can be argued that on physical grounds the main motivation for studying spectral properties is their close connection (e.g. via the RAGE-Theorem) with dynamical properties of the corresponding time evolution e−iHt , i.e. the propagation of wave packets under the time-dependent Schrödinger equation iψ  (t) = Hψ(t). However, the dynamical information following from spectral properties is not very accurate and examples have been found where spectral properties are misleading about the dynamics. In particular, this is the case for operators with singular continuous spectrum or dense point spectrum, spectral types quite common in quantum mechanical models of disordered media such as quasiperiodic or random Schrödinger operators, see for instance [20, 29, 53]. As a consequence, much of the recent work on hamiltonians governing disordered systems has focused on directly establishing dynamical properties. For example, it has been shown that Anderson-type random hamiltonians exhibit dynamical localization in various energy regimes, the property that wave packets ψ(t) stay localized in space for all times, see [1, 2, 23, 31, 32, 43], for example. The central object of interest in understanding dynamics is the unitary group U(t) = e−iHt , rather than the hamiltonian H itself. As short time fluctuations will generally not have a major impact on long time dynamics, one may discretize time by choosing a time unit T > 0 and study U(nT) = U n = e−inT H as n → ∞, with the fixed unitary propagator U = U(T). To further stress the role of the propagator as the central object in studies of dynamics, consider hamiltonians H(t) which depend periodically on time, H(t + T) = H(t) for some T > 0 and all t. In this case the large time behavior of solutions of iψ  (t) = H(t)ψ(t) is governed by the unitary propagator U(nT, 0) = U n , where U = U(T, 0) is commonly referred to as the monodromy operator of the time-periodic system. Note that in this case U does not have a meaningful representation of the form ei A any more. While

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such representations with selfadjoint operators A exist for abstract reasons, the operator A may have little to do with the time-dependent hamiltonian H(t). Actually, in the periodic or quasi-periodic cases, the operator A is linked to the so called quasienergy or Floquet operator [10, 37, 39, 57]. One consequence of this last fact is that it becomes legitimate to study time periodic systems by directly modeling the monodromy operator U based on physical properties. For example, time dependent analyses of electronic transport in disordered metallic rings have been considered within such a framework, [6–8, 11, 44]. Kicked systems, often used in the study of quantum chaos, provide another example of such models, see e.g. [12, 18, 21, 22, 24, 26, 45, 48]. Similar studies were performed on the dynamical properties of pulsed systems, given by smooth Floquet operators, in [9, 28, 38, 39]. Our central goal here is to investigate the dynamics of one such model U ω , which we call the unitary Anderson model, indicating the presence of disorder by the random parameter ω. The name is chosen by analogy to the selfadjoint Anderson model, which, in its discrete version on 2 (Zd ), takes the form hω = h0 + Vω .

(1.1)

The potential in (1.1) is given by real-valued i.i.d. random variables {Vω (k)}k∈Zd and h0 is a deterministic selfadjoint operator, most commonly the discrete Laplacian on Zd . By comparison, following our previous works [35, 40, 41], for the unitary Anderson model we choose a unitary operator on 2 (Zd ) of the form U ω = Dω S.

(1.2)

Here S is a deterministic unitary operator and Dω a multiplication operator by random phases, i.e. for every φ ∈ 2 (Zd ) and k ∈ Zd , ω

(Dω φ)(k) = e−iθk φ(k),

(1.3)

with i.i.d. random phases θkω taking values in T := R/2π Z. Note that, despite the formal analogy, there is no simple relation between selfadjoint and unitary Anderson models. In particular, due to non-commutativity, e−i(h0 +Vω ) is not a unitary Anderson model and, vice versa, Dω S can generally not be written in the form e−i(h0 +Vω ) . For S we choose what we consider to be a unitary analog of the discrete Laplacian. For d = 1 it has a five-diagonal structure which finds its roots in some physics models, see [13]. It turns out that S has the same five-diagonal structure as the so-called CMV-matrices, which have recently found much interest in the theory of orthogonal polynomials on the unit circle, where they arise as unitary analogs of Jacobi matrices, see [49, 50, 52] and references therein. For d > 1 we define S as a d-fold tensor product of its one-dimensional version. For details see Section 2. One of the reasons for this choice of S is that one can view the CMV-matrix structure as the simplest non-trivial band structure which a unitary operator on 2 (Z) can have, see e.g. [13], similar to the role of the discrete Laplacian among selfadjoint band matrices. Moreover,

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we choose S such that it will be invariant under translations by multiples of 2, yielding ergodicity of the unitary random operator U ω . Monodromy operators of the form (1.2), though not necessarily incorporating Anderson-type randomness, have also been proposed and studied in the physics literature [6, 11, 44] (see also [48]). A mathematical investigation of these models was initiated in [13] and continued in [27, 35, 36, 40, 41]. In particular, spectral localization for the unitary Anderson model was established in [35] for the one-dimensional model and in [41] for arbitrary dimension in the presence of large disorder. Here spectral localization refers to the property that U ω has pure point spectrum for almost every ω. We will provide proofs of dynamical localization (as formally defined in Section 3 below) for the unitary Anderson model in three different regimes: At arbitrary disorder and throughout the spectrum for the one-dimensional model as well as in the large disorder and band-edge regimes in arbitrary dimension (see Section 3 for a detailed description of these regimes). This coincides with the regimes where localization has been found to hold for selfadjoint Anderson models. Our approach to localization proofs will be via a unitary version of the fractional moment method, which was initiated as a tool in the theory of selfadjoint Anderson models by Aizenman and Molchanov in [4]. Dynamical localization will follow as a general consequence of exponential decay of spatial correlations in the fractional moments of Green’s function (Section 5). To complete the proof of dynamical localization in the three regimes described above, the latter property of Green’s function will then be established in those regimes. In fact, in the large disorder regime this has already been done in [41]. Its proof for the one-dimensional model is one of the main results of the thesis [34], from where we borrow the proofs presented here (Section 7). Some of our general results, in particular the proof that exponential decay of fractional moments of Green’s function implies dynamical localization (Section 5) and the proof that fractional moments of Green’s function are bounded (Section 4), are also essentially taken from [34]. The hardest, but possibly also most rewarding part of our work, is the proof of exponential decay of fractional moments of Green’s function in the band edge regime, which is carried out in Sections 8 to 14. Several preparatory sections are devoted to building up various mathematical tools which do not seem to be known in the context of unitary operators, such as the FeynmanHellmann theorem from perturbation theory (Section 9) and Combes-Thomas type bounds on eigenfunctions (Section 11). Along the way to localization we establish the spectral theoretic precursor of Lifshits tails of the integrated density of states for the unitary Anderson model (Section 12) as well as a decoupling procedure required in the iterative proof of exponential decay of fractional moments near the edges of the spectrum (Sections 10 and 13). In Section 6 we also include a proof of the general fact that, in the context of the unitary Anderson model, dynamical localization implies spectral localization, as previously known for selfadjoint Anderson models. In the unitary

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context, this follows from a version of the RAGE-Theorem provided in [30], whose proof we reproduce here. As already mentioned, when d = 1 and when restricted to l 2 (N) our unitary Anderson matrices U ω bear close resemblance with the CMV matrices in the theory of orthogonal polynomials on the unit circle, see [40]. These polynomials are determined by an infinite set of complex numbers on the unit disc that are called Verblunsky coefficients. Actually, U ω corresponds to a choice of Verblunsky coefficients characterized by constant moduli r and correlated random phases, see [35] for details. Other choices of random Verblunsky coefficients have been studied in the literature, see e.g. [51, 52, 55] and references therein. We note that for i.i.d. Verblunsky coefficients in the unit disc with rotation invariant distribution, Simon proves dynamical localization in [51]. While not spelled out explicitly, our results for the one dimensional case show that dynamical localization also holds for the CMV matrices considered in [35] with constant moduli Verblunsky coefficients and correlated phases. Finally, there is an underlying pedagogical goal to our paper: We use the unitary models considered here to give a self-contained presentation of the mathematical theory of Anderson localization via the fractional moment approach. Making use of state-of-the-art techniques from localization theory, we revisit the peculiarities of the one-dimensional case and techniques covering various regimes in the multi-dimensional case within a unitary framework. This requires developing and adapting all necessary background, which we do in a widely self-contained fashion.

2 The Unitary Anderson Model As the unitary Anderson model we denote a unitary random operator of the form U ω = Dω S

(2.1)

in 2 (Zd ). Motivated by earlier investigations in the physics literature, e.g. [11], this model was studied mathematically in [13, 35, 40, 41] and [36], from where we take the following definitions and basic results. A deterministic unitary operator S on l 2 (Zd ), sometimes referred to as the “free” unitary operator or “unitary Laplacian”, is constructed as follows: Starting with d = 1, let B1 and B2 be unitary 2 × 2 matrices given by     r t r −t B1 = and B2 = , (2.2) −t r t r with the real parameters t and r linked by r2 + t2 = 1 to ensure unitarity. Now let U e be the unitary matrix operator in l 2 (Z) found as the direct sum of identical B1 -blocks with blocks starting at even indices. Similarly, construct U o with identical B2 -blocks, where blocks start at odd indices. Define S0 = U e U o ,

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which will serve as the operator S in (2.1) for dimension d = 1. The operator S0 is unitary, with band structure ⎛ ⎞ .. 2 −t ⎜ . rt2 ⎟ ⎜ ⎟ r −rt ⎜ ⎟ 2 2 ⎜ ⎟ rt r rt −t ⎜ ⎟, S0 = ⎜ (2.3) 2 2 ⎟ −t −tr r −rt ⎜ ⎟ 2 ⎜ ⎟ rt r ⎝ ⎠ .. 2 . −t −tr where the position of the origin in Z is fixed by e2k−2 |Se2k  = −t2 , with ek (k ∈ Z) denoting the canonical basis vectors in l 2 (Z). Note also that S0 is invariant under translations by multiples of 2. Due to elementary unitary equivalences it will suffice to consider 0  t, r  1. Thus S0 is determined by t. We shall sometimes write S0 (t) to emphasize this parameter dependence. The spectrum of S0 (t) is given by the arc

 σ (S0 (t)) = (t) = eiϑ : ϑ ∈ [−λ0 , λ0 ] , (2.4) with λ0 := arccos(r2 − t2 ). The spectrum is symmetric about the real axis and grows from the single point {1} for t = 0 to the entire unit circle for t = 1. The spectrum is purely absolutely continuous for t > 0, e.g. Proposition 6.1 in [13]. To define the multidimensional unitary Laplacian, we follow [41] in viewing l 2 (Zd ) as ⊗dj=1l 2 (Z) so that for all k ∈ Zd , ek ek1 ⊗ ... ⊗ ekd . Using S0 = S0 (t) from above we define S = S(t) by S(t) = ⊗dj=1 S0 (t).

(2.5)

The spectrum of S(t) is



 σ (S(t)) = eiϑ : ϑ ∈ −dλ0 , dλ0 .

(2.6)

Throughout this paper | · | will denote the maximum norm on Z . Using this norm it is easy to see that S(t) inherits the band structure of S0 such that d

ek |S(t)el  = 0

if |k − l| > 2.

(2.7)

For the definition of the random phase matrix Dω in (2.1), introduce the d probability space ( , F , P), where = TZ (T = R/2π Z), F is the σ -algebra generated by cylinders of Borel sets, and P = k∈Zd μ, where μ is a non trivial probability measure on T. The expectation with respect to P will be denoted by E. We will assume throughout that μ is absolutely continuous with bounded density, dμ(θ) = τ (θ) dθ,

τ ∈ L∞ (T).

(2.8)

The random variables θk on ( , F , P) are defined by θk : → T, In other words, the distribution μ.

{θkω }k∈Zd

θkω = ωk ,

k ∈ Zd .

(2.9)

are T-valued i.i.d. random variables with common

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The diagonal operator Dω in 2 (Zd ) is given by ω

Dω ek = e−iθk ek .

(2.10)

With this choice for Dω and S = S(t) we define the unitary Anderson model U ω via (2.1). This definition and the periodicity of S ensures that the operator U ω is ergodic with respect to the 2-shift in . U ω also inherits the band structure of the original operator S. The general theory of ergodic operators, as for example presented in Chapter V of [17] for the self-adjoint case, carries over to the unitary setting. In particular, it follows that the spectrum of U ω is almost surely deterministic, i.e. there is a subset of the unit circle such that σ (U ω ) = for almost every ω. The same holds for the absolutely continuous, singular continuous and pure point parts of the spectrum: There are ac , sc and pp such that almost surely σac (U ω ) = ac , σsc (U ω ) = sc and σ pp (U ω ) = pp . Moreover, we can characterize in terms of the support of μ and of the spectrum of S; = exp (−i supp μ) σ (S) = {eiα : α ∈ [−dλ0 , dλ0 ] − supp μ}.

(2.11)

Here supp μ denotes the support of the probability measure μ, defined as supp μ := {a | μ((a − , a + )) > 0 for all  > 0}.

(2.12)

The identity (2.11) is shown in [40] for the one-dimensional model, but the argument carries over to arbitrary dimension. As t → 0, S0 (t) tends to the identity operator, whereas as t → 1, S0 (t) tends to a direct sum of shift operators. Accordingly, if t is zero then the operators U ω are diagonal and thus trivially have pure point spectrum. On the other hand, if t = 1 then it is not hard to see that all U ω are purely absolutely continuous (in fact, they are unitarily equivalent to a direct sum of shift operators). Thus, excluding the trivial special cases, we shall from now on assume that 0 < t < 1. For the unitary Anderson model the parameter t takes the role of a disorder parameter with small t corresponding to large disorder, as this means that U ω is dominated by its diagonal part.

3 The Results As discussed in the introduction, our main goal is to study regimes in which a quantum mechanical system governed by the unitary propagator U ω is dynamically localized. This can be expressed in terms of the transition amplitudes ek |U ωn el , whose squares measure the probability that a system initially in state el evolves into state ek after time n. By dynamical localization we will refer to the property that the expectation of these amplitudes stays exponentially small in the distance of k and l, uniformly for all times, i.e. the existence of constants C < ∞ and α > 0 such that   E sup |ek |U ωn el |  Ce−α|k−l| . (3.1) n∈Z

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In fact, what we will prove in several corresponding regimes is the stronger result that   E sup |ek | f (U ω )el |  Ce−α|k−l| , (3.2) f

where the supremum is taken over all functions f ∈ C(S) with f ∞ := supz∈S | f (z)|  1. Here S = {z ∈ C : |z| = 1} is the unit circle. Also, dynamical localization may only hold in an arc {eiθ : θ ∈ [a, b ]} of the ω spectrum of U ω . In this case, the spectral projection P[a,b ] of U ω onto this arc will be applied to the state el in (3.2), restricting the initial state to the localized part of the spectrum. Our detailed results, stated in the following two subsections, fall into two categories: We start with results which show that dynamical localization can be established by a unitary version of the fractional moments method, using as a criterion the exponential decay of fractional moments of spatial correlations of the Green function. We also show that dynamical localization generally implies spectral localization, i.e. that U ω almost surely has pure point spectrum in the corresponding part of the spectrum. Finally, we state that dynamical localization implies almost sure finiteness of all quantum moments of the position operator. All these results hold for arbitrary dimension d, for general values of the disorder parameter t ∈ (0, 1), and without restriction on the spectral parameter of the unitary operators U ω . Our second set of results concerns the proof of dynamical localization in three concrete regimes: for the one-dimensional unitary Anderson model, as well as for large disorder and at band edges in arbitrary dimensions. In each case this will be done by verifying exponential decay of the fractional moments of the Green function. 3.1 Fractional Moment Criteria for Localization For z ∈ C with |z| = 1 let G(z) = Gω (z) = (U ω − z)−1 ,

(3.3)

and G(k, l; z) = ek |G(z)el ,

k, l ∈ Zd

(3.4)

be the Green function of U ω (to use a term from the theory of selfadjoint hamiltonians in the unitary setting). The Green function becomes singular as z approaches the spectrum of U ω . The first insight which makes the fractional moment method a useful tool in localization theory is that these singularities are fractionally integrable with respect to the random parameters. This means that for s ∈ (0, 1) the fractional moments E(|G(k, l; z)|s ) have bounds which are uniform for z arbitrarily close to the spectrum. This is the content of our first result. In fact, we will need a somewhat stronger result later, namely that it suffices to average over the random variables θk and θl to get uniform bounds on |G(k, l; z)|s . The

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role of bounds of this type in localization proofs via the fractional moment method roughly corresponds to the use of Wegner estimates in the approach to localization by the method of multi scale analysis. Theorem 3.1 Assume that the random variables {θk }k∈Zd are i.i.d. with distribution μ satisfying (2.8). Then for every s ∈ (0, 1) there exists C(s) < ∞ such that  |G(k, l; z)|s dμ(θk )dμ(θl )  C(s) (3.5) for all z ∈ C, |z| = 1, all k, l ∈ Zd , and arbitrary values of θ j, j ∈ {k, l}. Consequently, E(|G(k, l; z)|s )  C(s),

(3.6)

for all z ∈ C, |z| = 1. The second statement simply derives from the bound (3.5), which uniform in the random variables θ j, j ∈ {k, l}, and the independence of the θ j. The proof of (3.5) is given in Section 4. In the next subsection we will identify several situations where the fractional moments E(|G(k, l; z)|s ) are not just uniformly bounded, but decay exponentially in the distance of k and l. The following general result shows that this can be used as a criterion for dynamical localization of U ω . Theorem 3.2 Assume that the random variables {θk }k∈Zd satisfy (2.8) and that for some s ∈ (0, 1), C < ∞, α > 0, ε > 0 and an interval [a, b ] ∈ T, E(|G(k, l; z)|s )  Ce−α|k−l|

(3.7)

for all k, l ∈ Zd and all z ∈ C such that 1 − ε < |z| < 1 and arg z ∈ [a, b ]. Then there exists C˜ such that ⎡ ⎤ ⎢ ⎥ ω ˜ −α|k−l|/4 E ⎣ sup |ek | f (U ω )P[a,b ] el |⎦  Ce f ∈C(S)

f ∞ 1

(3.8)

for all k, l ∈ Zd . Here, as usual arg z ∈ T refers to the polar representation z = |z| exp (i arg z) of a complex number. We prove Theorem 3.2 in Section 5. It has been shown in [41] that fractional moment bounds of the form (3.7) for a unitary Anderson model imply spectral localization via a spectral averaging technique, following an approach to localization which is due to Simon and Wolff [54] for the selfadjoint Anderson model. For completeness, we present a direct proof of the fact that dynamical localization expressed by (3.8) implies spectral localization in the unitary setup. This follows from a simple adaptation of arguments borrowed from Enss-Veselic [30], see also [15],

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on the geometric characterization of bound states, i.e. a RAGE-type theorem for unitary operators. We prove Proposition 3.1 Assume that for an interval [a, b ] there exist constants C < ∞ and α > 0 such that ⎡ ⎤ ⎢ ⎥ ω −α|k−l| E ⎣ sup |ek | f (U ω )P[a,b ] el |⎦  Ce

(3.9)

f ∈C(S)

f ∞ 1

for all k, l ∈ Zd . Then (a, b ) ∩ cont = ∅, where cont = sc ∪ ac . In other words, almost surely spectrum.

(3.10) ω P[a,b ]Uω

has pure point

Another direct consequence of the dynamical localization estimate (3.8), ω 2 d is that it prevents the spreading of wave packets from P[a,b ] l (Z ) under the discrete dynamics generated by U ω . This dynamical localization property is measured in terms of the boundedness in time of all quantum moments of p the position operator on the lattice. More precisely, for p > 0 let |X|e be the maximal multiplication operator such that |X|ep e j = | j|ep e j,

for j ∈ Zd ,

(3.11)

where | j|e denotes the Euclidean norm on Zd . We have Proposition 3.2 Assume that there exist C < ∞ and α > 0 such that ⎡ ⎤ ⎢ ⎥ ω −α|k−l| E ⎣ sup |ek | f (U ω )P[a,b ] el |⎦  Ce f ∈C(S)

f ∞ 1

(3.12)

for all k, l ∈ Zd . Then, for any p  0 and for any ψ in l 2 (Zd ) of compact support, ω sup |X|epU ωn P[a,b ] ψ < ∞ almost surely. n∈Z

(3.13)

Similar results hold under weaker support conditions on ψ. Our choice is made to keep things simple. The proofs of Propositions 3.1 and 3.2 are given in Section 6. 3.2 Localization Regimes The other main goal of our work is to identify three different regimes, where the fractional moment condition (3.7) can be verified, and thus dynamical localization follows by Theorem 3.2.

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3.2.1 Large Disorder As explained at the end of Section 2, the parameter t ∈ (0, 1) used in the definition of S(t) and thus, implicitly, also in the definition of U ω , can be thought of as a measure of the degree of disorder in the unitary Anderson model. Thus one can expect a tendency toward localization for small values of t. This was confirmed in [41] where the following was proven: Theorem 3.3 Suppose that the i.i.d. random variables {θk }k∈Zd have distribution μ satisfying (2.8) and let s ∈ (0, 1). Then there exists t0 > 0 and C < ∞ such that if t < t0 , there exists α > 0 so that E(|e j|U ω (U ω − z)−1 ek |s )  Ce−α| j−k|

(3.14)

for all j, k ∈ Zd and all z ∈ C with |z| = 1. In fact, [41] considers a more general model in which different parameters t i are chosen in each factor of (2.5) and shows (3.14) under the condition that i ti is sufficiently small. Using (4.1) below this implies that (3.7) holds for all |z| = 1. Thus in the large disorder regime t < t0 dynamical localization holds on the entire ω spectrum of U ω by Theorem 3.2 (the spectral projection P[a,b ] in (3.8) can be dropped).

3.2.2 The One-Dimensional Model For the one-dimensional self-adjoint Anderson model, localization holds throughout the spectrum, independent of the amount of disorder. The same is true for the unitary Anderson model, as implied by the following result: Theorem 3.4 Let d = 1 and suppose that the i.i.d. random variables {θk }k∈Z have distribution μ satisfying (2.8). Then for every t < 1 there exist s > 0, C < ∞ and α > 0 such that E(|G(k, l; z)|s )  Ce−α|k−l|

(3.15)

for all z ∈ C such that 0 < ||z| − 1| < 1/2 and all k, l ∈ Z . d

By Theorem 3.2 this implies dynamical localization for the one-dimensional unitary Anderson model throughout the spectrum. Many of the special tools which have been heavily exploited in studies of the one-dimensional self-adjoint Anderson model, are also available for the onedimensional Unitary model. First of all, there is a transfer-matrix formalism which allows the definition of Lyapunov exponents. In particular, it has been shown in [36] that under assumption (2.8) (in fact, for much more general distributions) the Lyapunov exponent is positive on the entire spectrum and continuous in the spectral parameter. This is the central ingredient into

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Lemma 7.1 stated in Section 7 below. For a proof of Lemma 7.1, which is a close analog of a result proven for the selfadjoint one-dimensional Anderson model in [16], we will refer to [34]. In Section 7 we will then explain in detail how this leads to (3.15). 3.2.3 Band Edge Localization For notational simplicity and without restriction we assume for the following result that supp μ ⊂ [−a, a] with a ∈ (0, π ) and −a, a ∈ supp μ. Furthermore, we assume that a + dλ0 < π,

(3.16)

which by (2.11) guarantees the existence of a gap in the almost sure spectrum of U ω ,

iϑ  e : ϑ ∈ (dλ0 + a, 2π − dλ0 − a) ∩ = ∅, (3.17) and that ei(dλ0 +a) and ei(2π −dλ0 −a) are band edges of . Our main result is Theorem 3.5 Assume (2.8) and let 0 < s < 1/3. There exist δ > 0, α > 0 and C < ∞ such that E(|G(k, l; z)|s )  Ce−α|k−l|

(3.18)

for all k, l ∈ Z and all z ∈ C with |z| = 1 and arg z ∈ [dλ0 + a − δ, dλ0 + a] ∪ [2π − dλ0 − a, 2π − dλ0 − a + δ]. d

Note that by Theorem 3.2 this implies dynamical localization for U ω near the edges dλ0 + a and 2π − dλ0 − a of its almost sure spectrum. The strategy of the proof of Theorem 3.5 is the following. We control the expectation value of fractional moments of the infinite volume Green function in terms of the expectation value of fractional moments of the Green function of a finite volume restriction of the operator. This requires addressing several distinct issues. The first one is the definition of an appropriate finite volume restriction. We restrict the problem to a finite but large box, by introducing appropriate boundary conditions in Section 8. Our choice of boundary conditions is governed by the fact that we need monotoncity properties which are similar to those of Neumann conditions in the selfadjoint case. The link between the infinite and finite volume resolvents is provided by a geometric resolvent estimate and a decoupling argument, similar to the self-adjoint case. Provided one has good estimates on the expectation value of the fractional moments of the finite volume Green function, this allows to lift such estimates to the fractional moments of the infinite volume resolvent by means of an iteration, for large but fixed size of the box, in a neighborhood of the band edges. This second step is addressed in Section 13. To get the sought for estimates on the resolvent of the finite volume restriction, in the band edge regime, we need to control the probability that

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the finite volume restriction of U ω has eigenvalues close to the band edge. Quantitatively, this amounts to showing that the probability of a small distance, algebraic in the inverse size of the box, between the eigenvalues of this finite volume restriction and the band edges is exponentially small, as the size of the box increases, see Proposition 12.1. This is an expression of the fact that the spectrum close to the band edges is very fluctuating which gives rise to Lifshits tails in the density of states. To prove this, we follow the self-adjoint route, see [56]. We first study the effect of Neumann boundary conditions on the spectrum of S in Section 8, and we make use of a unitary version of the Feynman-Hellmann formula, Proposition 9.1, to control this effect on the spectrum of the random operator U ω . Then, the Lifshits tail estimate together with a unitary version of the Combes-Thomas estimate, Proposition 11.1, allow to show that the expectation of the moments of the finite volume Green function is exponentially small in a power of the size of the box, Proposition 14.1.

4 Boundedness of Fractional Moments In this section we prove Theorem 3.1. As the bound (3.5) is trivial for |z| < 1/2, it suffices to consider |z|  1/2 and |z| = 1. Below we prefer to work with the modified resolvent (U ω + z)(U ω − z)−1 . Since (U ω − z)−1 =

1 (U ω + z)(U ω − z)−1 − I , 2z

(4.1)

it easy to see that the existence of 0 < C < ∞ for which 

|ek |(U ω + z)(U ω − z)−1 el |s dμ(θk ) dμ(θl )  C,

(4.2)

for all z with |z| = 1, all k, l ∈ Zd , and uniformly in θ j, j ∈ {k, l}, gives the required bound. Key to the proof of (4.2) is knowledge of the exact algebraic dependence of the Green function on the two parameters θk and θl . Similar formulas for rank one and rank two perturbations of the resolvents of unitary operators have been derived in [49], Section 4.5, from where we took guidance. We mostly focus on the proof of (4.2) for the case k = l. At the end of the proof we comment on the simpler case k = l, where (4.2) only requires averaging over one parameter. For k = l we borrow an idea from [3] and introduce the change of variables α = 12 (θk + θl ), β = 12 (θk − θl ). This will have the effect of essentially reducing (4.2) to averaging over the single parameter α (although this still corresponds to a rank two perturbation).

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Let A = {k, l} ⊂ Zd and define



α, j ∈ A 0, j ∈ / A ⎧ j=k ⎨ β, ξ j = −β, j = l ⎩ 0, j∈ / A  0, j∈ A  θ ωj = θ ωj , j ∈ / A ηj =

(4.3)

(4.4)

(4.5)

 by Next, we define the diagonal operators Dα , Dβ and D Dα e j = e−iη j e j,

Dβ e j = e−iξ j e j,

 j = e−iθˆ j e j. De

(4.6)

Using these definitions we can write U ω = Dα Vω ,

(4.7)

 In what follows we explore the relation with the unitary operator Vω = Dβ DS. between the modified resolvents of U ω and Vω . Let P A be the orthogonal projection onto the span of {Vω−1 e j : j ∈ A}. Using that {Vω−1 e j : j ∈ Zd } is an orthonormal basis of l 2 (Zd ), simple calculations show that (U ω − Vω )(I − P A ) = 0 and Vω−1 U ω = e−iα I on range P A . In particular, U ω − Vω = (U ω − Vω )P A is a finite rank operator. Therefore, U ω = Vω (I − P A ) + e−iα Vω P A .

(4.8)

z = For z ∈ C \ {0} with |z| = 1, let Fz = P A (U ω + z)(U ω − z) P A while F P A (Vω + z)(Vω − z)−1 P A , both viewed as operators on the range of P A (i.e. 2 × 2-matrices). We see that  

z + F z∗ = P A 2I − 2|z|2 (Vω − z)−1 (Vω − z)−1 ∗ P A . F (4.9) −1

z∗ is invertible and F z + F z∗ < 0 for |z| > 1. Therefore, z + F This shows that F z is a dissipative operator, i.e. an operator A such that (A − A∗ )/2i > 0. −i F z−1 is also a dissipative operator. In the case |z| < 1, we have that Similarly, −i F −1   i Fz , i Fz are dissipative. z and Fz . Following Section 4.5 of Next we explore the relation between F [49] we use the fact that (x + z)(x − z)−1 = 1 + 2z(x − z)−1 along with (4.8) to obtain z = −2zP A (Vω − z)−1 Vω P A (e−iα − 1)P A (U ω − z)−1 P A . Fz − F

(4.10)

z , this can be rewritten in the form Using the definitions of Fz and F    z e−iα − 1 (1 − Fz ). z = 1 1 + F Fz − F (4.11) 2 −iα

For α ∈ / {0, π }, let m(α) = i 1+e ∈ R. A straightforward calculation shows 1−e−iα that     z + m(α) −1 − i −i F z−1 − m−1 (α) −1 . Fz = −i −i F (4.12)

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z depends on β and the  Note that F θ j, but not on α. From the definitions of Fz and P A and the fact that Vω−1 U ω = e−iα I on span{Vω−1 e j : j ∈ {k, l}}, a simple calculation shows that ek |(U ω + z)(U ω − z)−1 el  = Vω−1 ek |Fz Vω−1 el . Therefore,



(4.13)

|ek |(U ω + z)(U ω − z)−1 el |s dμ(θk ) dμ(θl )   ||τ ||2∞

  τ 2∞

2π



2π



0

0

0 2π

|Vω−1 ek |Fz Vω−1 el |s dθk dθl

Fz s dθk dθl

0



 2||τ ||2∞

2π

π −π



2π



Fz s dα

dβ,

(4.14)

0

where we have changed to the variables α and β and slightly enlarged the integration domain into the rectangle 0  α  2π , −π  β  π . We split the α-integral according to (4.12),  2π  2π  2π     z + m(α) −1 s dα+ z−1 −m−1 (α) −1 s dα.

Fz s dα 

−i F

−i F 0

0

0

(4.15) Recalling that m(α) has singularities at α ∈ {0, 2π }, we make the change of variables x = m(α),  2π   z + m(α) −1 ||s dα || −i F 0



R

= lim

R→∞ −R

=2

 n∈Z

2

 n∈Z

n

x2

  2 z + x −1 ||s dx || −i F +1

n+1

x2

  1 z + x −1 ||s dx || −i F +1

1 (|n| − 1)2 + 1



n+1

  z + x −1 ||s dx. || −i F

(4.16)

n

We can now complete the proof of (4.2) by treating the cases |z| > 1 and z is dissipative and Lemma 4.1 shows |z| < 1 separately. If |z| > 1 then −i F boundedness of the integral on the right of (4.16), uniform in n, |z| > 1, β and  2π z + m(α))−1 ||s  C(s). The second term  θ j. Thus, after summation, 0 ||(−i F z−1 is on the right of (4.15) can be bounded in a similar way, using that −i F dissipative as well. Inserting these bounds into (4.14) makes the β-integration trivial and completes the proof of (4.2) for the case |z| > 1.

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z and −i F z−1 and anti-dissipative. As Lemma 4.1 obviIf |z| < 1, then −i F ously also holds for anti-dissipative matrices A, the proof of (4.2) goes through with the same argument. The proof of (4.2) for the case k = l is similar but simpler. We don’t need a change of variables, but directly work with one of the parameters θl instead of z α, leading to rank one perturbations. The objects corresponding to Fz and F become scalars and we only have to use the trivial scalar version of Lemma 4.1 to conclude. We finally provide an elementary proof of the following Lemma which was used above. A much more general result of this form is given as Lemma 3.1 in [3]. Lemma 4.1 For every s ∈ (0, 1) there exists C(s) < ∞ such that 

(A + xI)−1 s dx  C(s)

(4.17)

E

for every dissipative 2 × 2-matrix A and every unit interval E. Proof First observe that a general dissipative 2 × 2-matrix A is unitarily equivalent to an upper triangular dissipative matrix (choose as unitary transformation any matrix whose first column is given by a normalized eigenvector of A). Thus we may assume that   a11 a12 , (4.18) A= 0 a22 which implies

⎛

(A + xI)−1

1 ⎜ a11 + x =⎝ 0

−

⎞ a12 (a11 + x)(a22 + x) ⎟ . ⎠ 1 a22 + x

(4.19)

The bound (4.17) follows if we can establish a corresponding fractional integral bound for the absolute value of each entry of (4.19) separately. For the diagonal entries this is obvious. We bound the upper right entry of (4.19) by |a12 | a12  (a11 + x)(a22 + x) |Im ((a11 + x)(a22 + x))| = The positive matrix

1 x

Im a11 +Im a22 |a12 |

+

Im(a11 a22 )

.

(4.20)

|a12 |

⎛

⎞ 1 Im a a 11 12 1 ⎜ ⎟ 2i Im A = (A − A∗ ) = ⎝ 1 ⎠ 2i − a¯ 12 Im a22 2i

(4.21)

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has positive determinant, i.e. det Im A = Im a11 Im a22 − |a12 |2 /4. We thus get Im a11 + Im a22 a12

2



2Im a11 Im a22 1  . 2 |a12 | 2

The latter allows to conclude the required integral bound for (4.20).

(4.22)  

5 Dynamical Localization via Green’s Function Here we will prove Theorem 3.2, i.e. that exponential decay of fractional moments of Green’s function implies dynamical localization. Our proof uses an idea which in the context of selfadjoint Anderson models is due to Graf [33], namely that second moments of an Anderson model’s Green function can be bounded in terms of its fractional moments (including, however, a scalar factor which becomes singular as the spectral parameter approaches the spectrum). While the details of the proof are more involved than in the selfadjoint case, we find a bound of this form for unitary Anderson models in Section 5.1. Another tool we use is the integral formula (5.28), which expresses operator functions f (U) in terms of the resolvent of U. In Section 5.2 we provide a proof of this formula, which combines the spectral theorem for unitary operators with the representation of Borel measures on T by Poisson integrals. Equipped with these tools we complete the proof of dynamical localization in Section 5.3. 5.1 A Second Moment Estimate We start by a bound of second moments of Green’s function in terms of its fractional moments, which holds pointwise in the spectral parameter. Proposition 5.1 Assume that the {θkω }k∈Zd satisfy (2.8). Then for every s ∈ (0, 1) there exists C(s) < ∞ such that     E 1 − |z|2 |G(k, l; z)|2  C(s) E(|G(m, l; z)|s ) (5.1) |m−k|4

for all |z| < 1 and k, l ∈ Zd . Proof Throughout the proof we will assume z = 0. The bound (5.1) carries over to z = 0 by continuity. ω ω For δ ∈ T, let ηk = e−i(θk +δ) − e−iθk . Then define Dδω = Dω + ηk Pk , where Pk is the orthogonal projection into the span of ek . Let U ωδ = Dδω S. Using the resolvent identity we have  δ −1  −1 Uω − z − (U ω − z)−1 = − U ωδ − z ηk Pk S(U ω − z)−1 ,

(5.2)

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for all z ∈ C such that 0 < |z| < 1. Letting F(z) = S(U ω − z)−1 and Fδ (z) = S(U ωδ − z)−1 , the last equation takes the form Fδ (z) − F(z) = −Fδ (z)ηk Pk F(z).

(5.3)

Denoting F(i, j, z) = ei |F(z)e j it is easy to see that Fδ (z) − F(z) = −ηk

F(z)Pk F(z) . 1 + ηk F(k, k, z)

(5.4)

Therefore, for all l ∈ Zd Fδ (k, l, z) =

F(k, l, z) . 1 + ηk F(k, k, z)

(5.5)

On the other hand, we also have that |Fδ (k, l, z)|2 



|Fδ (k, y, z)|2

y∈Zd

!  −1 " δ −1 #∗ ∗ $ = ek |S U ωδ − z Uω − z S ek .

(5.6)

Since U ωδ is a unitary operator, the following identity holds   " −1 #∗ −1 1 −1 δ δ δ Uω − = Uω . Uω − z z¯ z¯

(5.7)

Thus, it follows that % &    δ −1 −1 1 −1 δ δ |Fδ (k, l, z)|  ek |S U ω − z Uω − Dω ek z¯ z¯ % &   ω  δ −1 −e−i(θk +δ) 1 −1 δ ek |S U ω − z = Uω − ek . z¯ z¯ 2

(5.8)

Again using the resolvent identity, we see that '     (  δ −1  δ −1 1 −1 z¯ 1 −1 δ δ Uω − z Uω − = 2 Uω − z − Uω − . z¯ |z| − 1 z¯

(5.9)

Hence, ω

e−i(θk +δ) |Fδ (k, l, z)|  1 − |z|2 2

'

!

% &(    δ −1 $ 1 −1 δ ek |S U ω −z ek − ek |S U ω − ek . (5.10) z¯

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1 From (5.7), the definition of U ωδ and the fact that (U ωδ − z)−1 = − [I − z U ωδ (U ωδ − z)−1 ], it follows that % &   ! " −1 #∗ $ 1 −1 ω δ ek |S U ω − ek = −z¯ ei(θk +δ) ek | U ωδ − z ek z¯ ) !  $* −1 ω ek |ek = −z¯ ei(θk +δ) 1 − U ωδ U ωδ − z ) * ω ω = ei(θk +δ) 1 − ei(θk +δ) Fδ (k, k, z) . (5.11) Therefore, we now obtain that



−i(θ ω +δ) 1 2 e k Fδ (k, k, z) − 1 2 1 − |z|

 1 ω = |Fδ (k, k, z)|2 − |ei(θk +δ) − Fδ (k, k, z)|2 , (5.12) 1 − |z|2

|Fδ (k, l, z)|2 

since |x − y|2 = |x|2 + |y|2 − 2[x¯ y]. Using (5.5), to rewrite Fδ (k, k, z) in terms of elements of F, along with the definition of ηk we get  + ω 1 |F(k, k, z)|2 − |eiθk − F(k, k, z)|2 2 |Fδ (k, l, z)|  . (5.13) 1 − |z|2 |1 + ηk F(k, k, z)|2 This inequality gives, in particular, that F(k, k, z) = 0. Therefore (1 − |z|2 )|Fδ (k, l, z)|2 

1 − |1 − eiθk F(k, k, z)−1 |2 . |ηk + F(k, k, z)−1 |2

(5.14)

Finally note that the last inequality allows us to conclude that |1 − eiθk F(k, k, z)−1 |  1. One can also use the fact that |Fδ (k, k, z)|  ||(U ωδ − z)−1 | 

1 , 1 − |z|

(5.15)

for all δ ∈ T, to get a different upper bound on (1 − |z|2 )|Fδ (k, l, z)|2 . Since (5.5) can be rewritten as Fδ (k, l, z) =

1 F(k, l, z) , ηk + F(k, k, z)−1 F(k, k, z)

(5.16)

it follows that 1 − |ηk + F(k, k, z)−1 |  |z|. Then by choosing δ such that e−iδ = 1−eiθk F(k,k,z)−1 , we see that |1−eiθk F(k,k,z)−1 | |1 − eiθk F(k, k, z)−1 |  |z|.

(5.17)

Using this along with (5.16) we obtain the following upper bound (1 − |z|2 )|Fδ (k, l, z)|2 

1 − |1 − eiθk F(k, k, z)−1 |2 |F(k, l, z)|2 . |ηk + F(k, k, z)−1 |2 |F(k, k, z)|2

(5.18)

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Combining the two estimates (5.14) and (5.18) and using that for 0 < s < 1 we have min(1, |x|2 )  |x|s , it follows that (1 − |z|2 )|Fδ (k, l, z)|2 

1 − |1 − eiθk F(k, k, z)−1 |2 |F(k, l, z)|s . |e−iδ − (1 − eiθk F(k, k, z)−1 )|2 |F(k, k, z)|s

(5.19)

Letting y = 1 − eiθk F(k, k, z)−1 , this can be rewritten as (1 − |z|2 )|Fδ (k, l, z)|2 

(1 − |y|2 )|1 − y|s |F(k, l, z)|s . |e−iδ − y|2

(5.20)

Since the expectations of F and Fδ are related by ,

E |F(k, l, z)|2 = E dμ(θk + δ)|Fδ (k, l, z)|2 ,

(5.21)

it follows that E[(1 − |z|2 )|F(k, l, z)|2 ] .



(1 − |y|2 )|1 − y|s  ||τ ||∞ E |F(k, l, z)|s sup dδ |e−iδ − y|2 {y∈C:|y|<1} 0 . /  2π (1 − |y|2 ) s s  2 ||τ ||∞ E |F(k, l, z)| sup dδ −iδ . |e − y|2 {y∈C:|y|<1} 0 2π

Next we evaluate the integral , −iδ  2π  2π (1 − |y|2 ) e +y dδ −iδ = dδ |e − y|2 e−iδ − y 0 0  2π , 2e−iδ −1 . dδ −iδ = e −y 0

/

(5.22)

(5.23)

The latter integral can be easily evaluated using Cauchy integral formula, by simply substituting z = e−iδ and gives  2π (1 − |y|2 ) dδ −iδ = 2π. (5.24) |e − y|2 0 Therefore, E[(1 − |z|2 )|F(k, l, z)|2 ]  2s+1 π ||τ ||∞ E[|F(k, l, z)|s ].

(5.25)

Since S is a unitary operator with band structure, we see that for s ∈ (0, 1) E[|F(k, l, z)|s ] = E[|S∗ ek |(U ω − z)−1 el |s ]   C1 (s) E[|em |(U ω − z)−1 el |s ]. |m−k|2

(5.26)

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401

Finally, in order to get the required bound of the second moment of elements of (U ω − z)−1 we use again that S is a unitary operator with band structure. Therefore, for k, l ∈ Zd E[(1 − |z|2 )|ek |(U ω − z)−1 el |2 ] = E[(1 − |z|2 )|Sek |S(U ω − z)−1 el |2 ]   C2 (s) E[(1 − |z|2 )|F(m, l, z)|2 ]. |m−k|2

(5.27)  

Combining (5.27), (5.25) and (5.26) gives (5.1). 5.2 An Integral Representation for f (U)

We will reduce bounds for f (U) to bounds on resolvents by the formula  −1  −1 −1  iθ  1 − r2 2π  f (U) = w − lim− U − reiθ U − re−iθ f e dθ (5.28) r→1 2π 0 for f ∈ C(S) and U a unitary operator. This is a simple consequence of the representation of Borel measures on T by Poisson integrals: Let ϕ ∈ H be normalized and consider the non negative spectral measure dμϕ on T such that   ϕ|Uϕ = eiα dϕ|E(α)ϕ = eiα dμϕ (α), (5.29) T

T

where E(·) denotes the spectral family of U. We can thus rewrite for 0  r < 1   !  −1  −1 −1 $ 1 − r2 1 − r2 ϕ| U − reiθ U − re−iθ ϕ = dμϕ (α) iα iθ 2 T |e − re |  1 − r2 = dμϕ (α) 2 T 1 + r − 2r cos(θ − α) = u(r, θ ),

(5.30)

which coincides with the Poisson integral of the measure dμϕ . As a function of z = x + iy = reiθ , with the standard abuses of notations, the RHS of (5.30) is non negative and harmonic in the open unit disc ([47] Sect. 11.17), but it is not bounded as r → 1− at the atoms of dμϕ . Now, Theorem 3.9.8 in [46] on the representation of non negative Borel measures on T says that for any f ∈ C(S) we have  2π  iθ dθ = lim u(r, θ) f (e ) f (eiα )dμϕ (α) ≡ ϕ| f (U)ϕ. (5.31) r→1− 0 2π T Considering the matrix element ϕ| · ψ for arbitrary ϕ and ψ in H, we get the equivalent result by polarization, which proves (5.28).

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If P[a,b ] denotes the spectral projector of U on the arc [a, b ] ⊂ T and ϕ ∈ H is normalized, then dϕ|P[a,b ] E(α)ϕ ≡ dμϕ[a,b ] (α) = dμϕ (α)|[a,b ] .

(5.32)

By the same argument with P[a,b ] ϕ in place of ϕ, i.e. with dμϕ[a,b ] in place of dμϕ , we have for any f ∈ C(S) 1 − r2 P[a,b ] f (U) = w − lim− r→1 2π



b

 −1  −1 −1  iθ  U − reiθ U − re−iθ f e dθ

a

≡ w − lim− fr (U).

(5.33)

r→1

5.3 Conclusion of the Proof of Theorem 3.2 Let f ∈ C(S) such that f ∞  1 and consider 

1 − r2 fr (U) = 2π

b

 −1  −1 −1  iθ  U − reiθ U − re−iθ f e dθ.

(5.34)

a

We compute |ek | fr (U)e j|  1 − r2 b  |ek |(U − reiθ )−1 (U −1 − re−iθ )−1 e j|| f (eiθ )|dθ 2π a  1 − r2 b   |ek |(U − reiθ )−1 el ||e j|(U − reiθ )−1 el |dθ, 2π a d

(5.35)

l∈Z

which is independent of f . Hence, using Fatou’s Lemma and Fubini’s Theorem and taking the supremum over all f ∈ C(S) such that f ∞  1, /

. E sup |ek |P[a,b ] f (U ω )e j| f

/

.

= E sup lim− |ek | fr (U)e j| .

f r→1

/

 E lim inf sup |ek | fr (U)e j| − r→1

 lim inf − r→1

f ∞ 1

1 − r2 2π

 a

b



E[|ek |(U − reiθ )−1 el ||e j|(U − reiθ )−1 el |]|dθ.

l∈Zd

(5.36)

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403

By the Cauchy-Schwarz inequality and the second moment bound (5.1), we have E[|(1 − r2 )ek |(U − reiθ )−1 el ||e j|(U − reiθ )−1 el |]  (E[|(1 − r2 )ek |(U − reiθ )−1 el |2 ])1/2 (E[|(1 − r2 )e j|(U − reiθ )−1 el |2 ])1/2 ⎞1/2 ⎛ ⎞1/2 ⎛    C1 ⎝ E(G(m, l; reiθ )|s )⎠ ⎝ E(G(m, l; reiθ )|s )⎠ m:|m− j|4

m:|m−k|4

 C2 e−α|k−l|/2 e−α| j−l|/2 ,

(5.37)

where ultimately the assumption (3.7) on exponential decay of fractional moments of Green’s function was used. Since there exists a finite c such that  e−α(|k−l|+| j−l|)/2  ce−α|k− j|/4 (5.38) l∈Zd

we finally get

/

.

˜ −α|k− j|/4 , E sup |ek |P[a,b ] f (U ω )e j|  Ce

(5.39)

f

with C˜ = c C2 . This completes the proof of Theorem 3.2. 6 Dynamical Localization Implies Spectral Localization We start by recalling the geometrical concepts and results introduced in [30] in a general framework. Let U be a unitary operator in a separable Hilbert space H and let H pp (U), respectively Hc (U) denote the pure point, respectively continuous, spectral subspace of U. We will denote the spectral resolution of U by EU . A practical characterization of the vectors of H pp (U) and Hc (U) in terms of their behaviour under the discrete dynamics with respect to families of finite dimensional subspaces of H is provided in [30] and reads as follows: Let P = {Pr }r0 be a family of projections on H such that Pr = Pr2 = Pr∗ ,

Rank Pr < ∞,

and s − lim Pr = I. r→∞

(6.1)

We define the set of bounded trajectories with respect to the family P by  + b n MU (P) = ψ ∈ H | lim sup (I − Pr )U ψ = 0 . (6.2) r→∞ n∈Z

Similarly, the set of propagating trajectories with respect to P is defined by N  1

Pr U n ψ = 0, ∀r  0}, N→∞ 2N + 1 n=−N

p

MU (P) = {ψ ∈ H | lim p

b Both MU (P) and MU (P) are easily seen to be closed subspaces of H.

(6.3)

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Then the following holds. Lemma 6.1 For any unitary operator U on a separable Hilbert space H, and any family P as in (6.1), b MU (P) = H pp (U)

p

and MU (P) = Hc (U)

(6.4)

While the definition above suffices for our purpose, more general families of operators than P can be considered, see [30]. For completeness, we provide a detailed proof of this Lemma, which is a slight adaptation of the argument in [30]. Proof of Lemma 6.1 in three steps: b First, H pp (U) ⊂ MU (P): If Uϕ = eiα ϕ, then

(I − Pr )U n ϕ = (I − Pr )ϕ → 0, uniformly in n as r → ∞. (6.5) We conclude using that H pp (U) = {ϕ | Uϕ = eiα ϕ} and the fact that b MU (P) is closed. p b Second, MU (P) ⊥ MU (P): p b Let ϕ ∈ MU (P) and ψ ∈ MU (P). Then ϕ|ψ =

N N   1 1 ϕ|ψ = U n ϕ|U n ψ 2N + 1 2N + 1 n=−N

=

n=−N

N 

1 U n ϕ|(I − Pr )U n ψ + U n ϕ|Pr U n ψ. (6.6) 2N + 1 n=−N

By our choice of ψ, for any  > 0 there exists r(), uniform in n, such that |U n ϕ|(I − Pr() )U n ψ| < . And our choice of ϕ and the selfadjointness of Pr() imply N  1 U n ϕ|Pr() U n ψ 2N + 1 n=−N



N  1

Pr() U n ϕ

ψ → 0 as N → ∞, 2N + 1 n=−N

(6.7)

which shows that ϕ|ψ = 0. p Third, Hc (U) ⊂ MU (P): Let ψ ∈ Hc (U) = Pc (U)H, where Pc (U) is the spectral projector of U on the continuous spectral subspace of U. Since the orthogonal projector Pr is finite dimensional for any r, there exists vectors ϕi ∈

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405

Ran Pr such that we can write for some m < ∞ Pr =

m 

|ϕi ϕi |,

ϕi |ϕ j = δij.

where

(6.8)

i=1

Hence, Pr U n ψ 2 = rem we have

m i=1

|ϕi |U n ψ|2 . Now, by the Spectral Theo

fϕi ,ψ (n) := ϕi |U ψ = n

T

eixn dμϕi ,ψ (x),

(6.9)

where dμϕ,ψ (x) = dPc (U)ϕi |EU (x)Pc (U)ψ is a continuous complex valued measure. Thus, by Wiener’s Theorem, N   1 | fϕi ,ψ (n)|2 = |μϕi ,ψ ({x})|2 = 0. N→∞ 2N + 1 x∈T n=−N

lim

(6.10)

Consequently, making use of Cauchy-Schwarz, N  1

Pr U n ψ  2N + 1 n=−N

'

' =

N  1

Pr U n ψ 2 2N + 1 n=−N

m  i=1

(1/2

N  1 | fϕi ,ψ (n)|2 2N + 1

(1/2 (6.11)

n=−N

which tends to zero as N → 0. Finally the decomposition H = H pp (U) ⊕ Hc (U) leads to the conclusion.   Proof of Proposition 3.1 Coming back to the random situation at hand, we construct a suitable family of projectors P = {Pr }r0 by means of Pr =



|ek ek |, r  0,

(6.12)

k∈Zd |k|r ω d and consider the vectors of the form P[a,b ] e j , j ∈ Z . We have for all n ∈ Z

ω

(I − Pr )U ωn P[a,b ] e j =

⎧ ⎪ ⎪ ⎨



⎪ ⎪ ⎩k∈Zd

|k|>r

ω n 2 |ek |P[a,b ] U ω e j |

⎫1/2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(6.13)

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ω n Therefore, since |ek |P[a,b ] U ω e j |  1,

ω sup (I − Pr )U ωn P[a,b ] e j = n



⎧ ⎪ ⎪ ⎨ sup

⎪ ⎪ ⎩

n

sup n

ω n 2 |ek |P[a,b ] U ω e j |

k∈Zd |k|>r

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩





ω n |ek |P[a,b ] U ω e j |

k∈Z |k|>r d

⎫1/2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎫1/2 ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(6.14)

Thus, by Fatou’s Lemma and Cauchy-Schwarz inequality,   n ω E lim sup (I − Pr )U ω P[a,b ] e j

r→∞ n

⎛⎧ ⎨ ⎜  lim inf E ⎝ sup r→∞ ⎩ n

 k∈Zd |k|>r

⎫1/2 ⎞ ⎬ ⎟ ω n |ek |P[a,b ⎠ ] U ω e j | ⎭

⎧ ⎛ ⎞⎫1/2 ⎪ ⎪ ⎪ ⎪ ⎨ ⎜  ⎟⎬ ω n ⎜ ⎟  lim inf E ⎝sup |ek |P[a,b ] U ω e j|⎠ . r→∞ ⎪ ⎪ n ⎪ ⎪ ⎭ ⎩ k∈Zd

(6.15)

|k|>r

Now, by Fubini’s Theorem, ⎛ ⎞

⎛

⎞

 ⎜ ⎟ ⎜ ⎟ ω n ω n ⎟ ⎜ ⎟ E⎜ |ek |P[a,b sup |ek |P[a,b ] U ω e j |⎠  E ⎝ ] U ω e j |⎠ ⎝sup n n k∈Zd |k|>r

k∈Zd |k|>r

=





E

k∈Zd |k|>r

 C˜



ω n sup |ek |P[a,b ] U ω e j | n

˜ α| j|/4 e−α|k− j|/4  Ce

k∈Z |k|>r d





e−α|k|/4 ,

(6.16)

k∈Z |k|>r d

where the right hand side decays exponentially fast to zero as r → ∞. As a consequence,   n ω E lim sup (I − Pr )U ω P[a,b ] e j = 0, (6.17) r→∞ n

so that there exists a set j ⊂ such that P( j) = 1 and for all ω ∈ j ω lim sup (I − Pr )U ωn P[a,b ] e j = 0.

r→∞ n

(6.18)

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ω ˜ In other words, P[a,b ] e j ∈ H pp (U ω ), ∀ω ∈ j . Then, = ∩ j∈Zd j is a set of ˜ Pω e j ∈ H pp (U ω ), ∀ j ∈ Zd . Hence, probability one such that for all ω ∈ , [a,b ] ω 2 d P[a,b ] l (Z ) ⊂ H pp (U ω ), a.s.

(6.19)  

 Proof of Proposition 3.2 By assumption, ψ = k ψk ek satisfies ψk = 0 if |k| > R, for some R > 0. Hence, by Cauchy-Schwarz ω 2

|X|epU ωn P[a,b ] ψ =



ω 2 |e j||X|epU ωn P[a,b ] ψ|

j

=



| j|2e p

j



  j



2

k ω 2 2 | j|2e p |e j|U ωn P[a,b ] ek | ψ

|k| R

  j

 ω e j|U ωn P[a,b ] ek ψk

ω 2 | j|2e p |e j|U ωn P[a,b ] ek | ψ ,

(6.20)

|k| R

ω since |e j|U ωn P[a,b ] ek |  1. By the same steps as those performed in the previous proof, one gets that

     ω 2p n ω E sup |X|epU ωn P[a,b ψ

< ∞ if | j | E |e |U P e | < ∞. sup j e ω [a,b ] k ] n∈Zd

j |k| R

n

(6.21) That the latter sum is finite follows from (3.8), which ends the proof.

 

7 One-Dimensional Localization In this section we prove Theorem 3.4. 7.1 Basic Properties of the One-Dimensional Model The proof of Theorem 3.4 uses a number of tools, which are specific to the one-dimensional model, where the operators studied here can be considered as unitary analogs of Jacobi matrices. We start this section by briefly presenting the properties which we will need. While we have a somewhat different point of view, much of this is related or equivalent to facts on CMV-matrices, which can be found throughout [49, 50].

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Due to the specific band structure of U ω , the generalized eigenvectors can be studied using complex 2 × 2-transfer matrices. Specifically, generalized eigenvectors are solutions (not necessarily in l 2 ) of the eigenvalue equation  U ω ψ = zψ, ψ= ψk ek , (7.1) k∈Z

for z ∈ C\{0} and characterized by the relations     ψ2k+1 ψ2k−1 ω ω = Tz (θ2k , θ2k+1 ) , ψ2k+2 ψ2k

(7.2)

for all k ∈ Z. Here the transfer matrices Tz : T2 → GL(2, C) are defined by   ⎞ ⎛ r i(θ−η) e−iη e−iη − − e ⎟ ⎜ z t z    ⎟ . Tz (θ, η) = ⎜ (7.3) −iη iθ 2 −iη ⎝r ⎠ e ze r e 1− − 2 + 2 1 + ei(θ −η) − t z t t z ω

ω

ω ω Note that det Tz (θ2k , θ2k+1 ) = ei(θ2k −θ2k+1 ) has modulus one and is independent of z. We have for any n ∈ N       ψ2n−1 ψ−1 ψ−1 ω ω ω ω = Tz (θ2(n−1) , θ2(n−1)+1 ) · · · Tz (θ0 , θ1 ) ≡ Tz (ω, n) , (7.4) ψ2n ψ0 ψ0



ψ−2n−1 ψ−2n



ω ω ω ω −1 = Tz (θ−2n , θ−2n+1 )−1 · · · Tz (θ−2 , θ−1 )



ψ−1 ψ0



 ≡ Tz (ω, −n)

 ψ−1 . ψ0 (7.5)

We also set Tz (ω, 0) = I. The transfer matrix formalism allows to introduce the Lyapunov exponent γ (z); γ (z) = lim

n→∞

E(ln ||Tz (ω, n)||) . n

(7.6)

Positivity and continuity of the Lyapunov exponent for all values of z under the current assumptions was proven in [36]. A consequence of these properties of γ (z) is the following unitary version of Lemma 5.1 of [16]. Lemma 7.1 Assume that the random variables {θk }k∈Zd satisfy (2.8), then for each compact subset  of C there exist α = α() > 0 and 0 < δ = δ() < 1 and C = C() < ∞ such that E[||Tz (ω, n)v||−δ ]  Ce−αn

(7.7)

for all z ∈ , n  0 and unit vector v ∈ C2 . We omit the proof of this lemma which is very similar to the one given for the self-adjoint Anderson model in [16], see Appendix A in [34] for details.

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We will frequently work with restrictions of the infinite matrices U ω to discrete finite intervals [a, b ] or half-lines [a, ∞) and (−∞, b ], respectively. Here we slightly abuse notation and write, for example, [a, b ] for [a, b ] ∩ Z. To guarantee that the restrictions remain unitary we have to choose suitable boundary conditions. At each finite endpoint these boundary conditions can be labeled by a parameter in T. For a ∈ Z and η ∈ T, the unitary operator Sη[a,∞) on l 2 ([a, ∞)) is constructed as follows: If a = 2n is even, let the 2 × 2 matrices B1 and B2 be defined as in (2.2), then let U e[2n,∞) be the unitary operator in l 2 ([2n, ∞)) found as the direct sum of identical B1 -blocks with blocks starting at 2n. On the other hand construct U o[2n,∞) starting with a single 1 × 1 block eiη , then identical B2 -blocks = U e[2n,∞) U o[2n,∞) . The operator S[2n,∞) on starting at 2n + 1. Now let S[2n,∞) η η 2 l ([2n, ∞)) will have a band structure ⎛

S[2n,∞) η

reiη rt ⎜−teiη r2 ⎜ ⎜ rt ⎜ 2 =⎜ −t ⎜ ⎜ ⎝

⎞

−t2 −rt r2 rt −tr r2 rt

−t2 −rt r2

−t2

−tr

..

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(7.8)

.

The parameter η can be thought of as a boundary condition at 2n. Similarly, U e(−∞,2n+1] is found as the direct sum of identical B1 -blocks with blocks starting at even indices, while U o(−∞,2n+1] has identical B2 -blocks starting at odd indices, with (U o(−∞,2n+1] )(2n + 1, 2n + 1) = eiη . Thus, ⎛

S(−∞,2n+1] η

⎞

..

⎜ . ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

rt r2 rt −t2

− t2 −rt r2 −rt

rt r2 rt −t2

⎟ ⎟ ⎟ ⎟ −t2 ⎟. ⎟ −rt ⎟ r2 teiη ⎠ −rt reiη

(7.9)

we slightly modify this construction, this To define Sη(−∞,2n] and S[2n+1,∞) η time filling up U e[2n+1,∞) and U e(−∞,2n] with a 1 × 1-block eiη , respectively, yielding ⎛ Sη(−∞,2n]

..

⎜ . ⎜ ⎜ =⎜ ⎜ ⎝

⎞ rt r2 rt −t2

−t2 −rt r2 −rt

rt r2 teiη

⎟ ⎟ ⎟ , −t2 ⎟ ⎟ −rt⎠ reiη

(7.10)

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while

S[2n+1,∞) η

⎛ iη re ⎜ rt ⎜ 2 ⎜ = ⎜−t ⎜ ⎝

⎞

−teiη r2 rt −rt r2 rt

−t2 −rt r2

−t2

−rt

..

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

(7.11)

.

In similar fashion, for integers −∞ < a < b < ∞ we can construct the ] unitary operator Sη[a,b with ηa boundary condition at a and ηb boundary a ,ηb condition at b , for example we have ⎛ iη ⎞ re 2n rt −t2 ⎜−teiη2n r2 ⎟ −rt ⎜ ⎟ 2 2 ⎜ ⎟ rt −t rt r ⎜ ⎟ [2n,2m] 2 2 Sη2n ,η2m = ⎜ (7.12) ⎟. −t −rt r −rt ⎜ ⎟ ⎜ ⎟ . .. ⎝ ⎠ iη2m reiη2m te Finally, we define [a,b ] ] = Dω[a,b ] Sη[a,b , U ω,η a ,ηb a ,ηb

(7.13)

where the diagonal operator Dω[a,b ] on l 2 ([a, b ]) is defined as in (2.10). Similarly, [a,∞) (−∞,b ] and U ω,η . we define U ω,η [a,b ] [a,∞) (−∞,b ] As before, the generalized eigenvectors of U ω,η , U ω,η and U ω,η a ,ηb are characterized by the relations (7.2) are supplemented with appropriate relations to reflect the boundary conditions, see [34] for details. In the proof of [a,b ] = Theorem 3.4 we only use η = 0, so for the rest of this section we write U ω,0,0 [a,∞) (−∞,b ] [a,b ] [a,∞) (−∞,b ] U , U ω,0 = U and U ω,0 =U for simplicity, also frequently leaving the ω-dependence implicit. Other boundary conditions will be used later, see Section 8. In the following discussion we use the notation U [a,b ] for general −∞  a < b  ∞, i.e. we write U [−∞,∞] , U [a,∞] and U [−∞,b ] for U, U [a,∞) and U (−∞,b ] . Another feature of the one-dimensional model is that Green’s function G(k, l; z) can be expressed in terms of two generalized eigenfunctions to z which, separately at each endpoint a, b , are square-summable or satisfy the boundary condition at the endpoint. For a solution ϕ of (U − z)ϕ = 0, we define 4 ϕ by   2   4 ϕ2n t ϕ2n−1 rt = . (7.14) 4 ϕ2n+1 ϕ2n rt r2 − zeiθ2n A straightforward calculation shows that 4 ϕ is characterized by the relations     4 ϕ2k 4 ϕ2k−2 5 = Tz (θ2k−1 , θ2k ) , (7.15) 4 ϕ2k+1 4 ϕ2k−1

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for all k ∈ Z. Here the transfer matrices 5 Tz : T2 → GL(2, C) are defined by 5 Tz (θ, η) = Tzt (η, θ),

(7.16)

where Tz (η, θ) is given by (7.3) and T t denotes the transpose of T. For −∞  a < b  ∞ let G[a,b ] (z) = (U [a,b ] − z)−1 .

(7.17)

To any z not in the spectrum of U [a,b ] choose generalized eigenvectors ϕ a and ϕ b as follows: If a is finite, then ϕ a is the unique solution of (U [a,∞) −z)ϕ a = 0 with ϕ a (a) = 1, i.e. a generalized eigenvector to z which satisfies the boundary condition at a. If a = −∞, then for ϕ a we choose a non-trivial solution of (U − z)ϕ a = 0, which is square-summable at −∞. In the latter case ϕ a is determined up to a constant (one can construct if from the tail of (U − z)−1 e0 at −∞ and the fact that the transfer matrices (7.3) have determinant of modulus one shows that there can’t be two linearly independent solutions which are square-summable at −∞). Similar we choose ϕ b with prescribed boundary behavior at b . The following proposition gives an expression of the elements of G[a,b ] (z) ϕa, 4 ϕ b defined as in (7.14). in terms of ϕ a and ϕ b and the corresponding 4 Proposition 7.1 For all finite k, l with a  k, l  b , if l = 2n or l = 2n + 1, ' ϕlb ϕka if k < l or k = l are even, cl 4 (7.18) G[a,b ] (k, l; z) = cl 4 ϕla ϕkb if k > l or k = l are odd, where cl =

eiθl . a b a b 4 ϕ2n+1 4 ϕ2n −4 ϕ2n 4 ϕ2n+1

Proof A straightforward, if rather tedious, calculation shows that the matrix whose entries are given by the right hand side of (7.18) is indeed the inverse of   U ω[a,b ] − z, see [34] for details. We conclude this section by proving the following lemma that is used later in the proof of Theorem 3.4. Lemma 7.2 For z ∈ C with 0 < ||z| − 1| < 1/2, a ∈ Z and s ∈ (0, 1) there exists 0 < Cμ (t, s) < ∞ such that  2π 6  a  6−s 1 6 ϕ2m−1 6 dμ(θ2m ) a  C (t, s) (7.19) 6 6 , μ a a s ϕ2m |tϕ2m−1 + (r − zeiθ2m )ϕ2m | 0 for all m  a + 2. a a Proof First note that both ϕ2m−1 and ϕ2m are independent of θ2m and can not vanish simultaneously. This follows from the fact that the transfer matrices needed to construct them via (7.2) from ϕ a (a) = 1 only contain θa , . . . , θ2m−1 .

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Therefore we have the following cases: Case 1

a ϕ2m

6 a 6 6 6 ϕ a 2m−1 6  2|ϕ a = 0, using that 6 2m−1 | + 2|ϕ2m |, the bound follows 6 6 ϕa 2m

directly. a Case 2 ϕ2m = 0. In this case  2π dμ(θ2m ) 0

=



1 a s |ϕ2m |

1 a a s |tϕ2m−1 + (r − zeiθ2m )ϕ2m | 2π

dμ(θ2m )

0

1 ϕa |t ϕ2m−1 a 2m

+ (r − zeiθ2m )|s

.

(7.20)

Let M = sup{|r − zeiθ2m | : θ ∈ [0, 2π ], 0 < |z| − 1 < 1/2} < ∞ and distinguish between two subcases; Case 2a If t|

a ϕ2m−1 | a ϕ2m

> 2M, it follows that t

On the other hand

a a ϕ2m−1 t ϕ2m−1 + (r − zeiθ2m ) > . a a ϕ2m 2 ϕ2m

 6  a  6s  1 s a 6 ϕ2m−1 6 2 +  |ϕ2m−1 |s . 6 6 a ϕ2m M

(7.21)

(7.22)

Using these estimates we conclude that    a 6  2π 1 1 s6 2s+1 π ||τ ||∞ 6 ϕ2m−1 6−s dμ(θ2m ) a  2+ 6 . 6 a a s iθ s 2m ϕ2m |tϕ2m−1 +(r − ze )ϕ2m | t M 0 (7.23) ϕa

Case 2b Assume that t| ϕ2m−1 |  2M. For all 0 < s < 1 there exists 0 < Cμ (s) < a 2m ∞ such that for all β ∈ C  1 dμ(θ) ±iθ  Cμ (s), (7.24) |(e − β)|s T see e.g. [41]. Using this we get the bound  2π 2s Cμ(1) (s) 1 dμ(θ2m ) a  a s a s . |tϕ2m−1 + (r − zeiθ2m )ϕ2m | |ϕ2m | 0 However, under the current assumption we have s 6  a  6s  M 6 ϕ2m−1 6 a s  4 |ϕ2m |, + 2 6 6 a ϕ2m t which gives the required bound.

(7.25)

(7.26)  

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7.2 Proof of Theorem 3.4 We now turn to the proof of Theorem 3.4. The proof is done in three steps: first we prove that it suffices to deal with even matrix elements of Green’s function, i.e. that we only need to show (3.15) for the case that k = 2n, l = 2m. This mainly serves the purpose of avoiding to cover four separate sub-cases. Then we show that proving the bound (3.15) for an element (2n, 2m) of (U ω − z)−1 can be reduced to proving the same bound for the element (2n, 2m) of the resolvent of the finite volume operator U ω[2n,2m] at z. Finally, we prove that expectation of fractional moments of G[2n,2m] (2n, 2m; z) decays exponentially. Based on the Green’s function formula (7.18) this will be found by combining Lemma 7.2 with Lemma 7.1. Step One The following Lemma shows that the expectation of a fractional moment of any element G(k, l; z) can be reduced to the expectation of fractional moments of even matrix elements of G(z). This comes at the cost of enlarging the fractional exponent s due to the use of Hölder’s inequality. Lemma 7.3 Let s ∈ (0, 1/4) and k, l ∈ Z such that |k − l| > 4. Choose n, m ∈ Z such that k ∈ {2n, 2n + 1}, l ∈ {2m, 2m + 1}. There exists κ(t, s, μ) < ∞ such that E[|G(k, l; z)| ]  κ(t, s, μ) s

1  

1/4

E[|G(2n + 2i, 2m − 2 j; z)|4s ]

,

(7.27)

i, j=0

for all z ∈ C with 0 < ||z| − 1| < 1/2. Proof Using the definition of n, m and that |k − l| > 4 one has |n − m|  2. Since 4 ϕ ±∞ , defined by (7.14), satisfies (7.15) and (7.16), a straightforward calculation shows that ±∞ 4 ϕ2m+1 =

1 ω iθ2m−1

rt(e − 1/z) ,   + 1 ω ω ω ω ω ±∞ ±∞ × r2 eiθ2m +eiθ2m−1 − −zei(θ2m +θ2m−1 ) 4 ϕ2m − t2 eiθ2m 4 ϕ2m−2 . (7.28) z

Using Theorem 7.1 along with (7.15) it follows that for k ∈ / {2m − 1, 2m} ω

G(k, 2m + 1; z) =

eiθ2m+1 ω iθ2m−1 rt(e − 1/z) ,  ω  e−iθ2m ω ω ω × r2 1 + e−i(θ2m −θ2m−1 ) − − zeiθ2m−1 z ω ω 2 i(θ2m−1 −θ2m−2 )

×G(k, 2m; z) − t e

+ G(k, 2m − 2; z) .

(7.29)

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By Hölder’s inequality and (7.24) it follows that for s ∈ (0, 1/4) there exists 0 < Cμ(1) (s, r) < ∞ such that E[|G(k, 2m + 1; z)|s ]    Cμ(1) (s, r) (E[|G(k, 2m; z)|2s ])1/2 + (E[|G(k, 2m − 2; z)|2s ])1/2 .

(7.30)

Similarly, using (7.2) and (7.3) we obtain ±∞ ϕ2n+1

−t = ω iθ2n+1 r(e − 1/z)



+ 1 ±∞ ω ±∞ iθ2n ϕ + e ϕ2n . z 2n+2

(7.31)

Thus, for l ∈ / {2n, 2n + 1}, s ∈ (0, 1/2) and all z ∈ C with 0 < ||z| − 1| < 1/2, E[|G(2n + 1, l; z)|s ]    Cμ(2) (s, r) (E[|G(2n + 2, l; z)|2s ])1/2 + (E[|G(2n, l; z)|2s ])1/2 .

(7.32)

It readily follows from (7.30) and (7.32) that for |n − m| ∈ / {0, 1} and all s ∈ (0, 1/4) E[|G(2n + 1, 2m + 1; z)|s ]  κ˜ μ(1) (s, r)

1  

1/4

E[|G(2n + 2i, 2m − 2 j; z)|4s ]

.

i, j=0

(7.33) This proves the Lemma for the case k = 2n + 1, l = 2m + 1. The other cases are more direct.   Step Two Let |[k, l]| denote the interval [min{k, l}, max{k, l}]. In what follows we show that the expectation of fractional moments of G(2n, 2m; z) can be reduced to that of G|[2n,2m]| (2n, 2m; z). Lemma 7.4 For s ∈ (0, 1/3) and n, m ∈ Z with |n − m|  2, we have E[|G(2n, 2m; z)|s ]  Cμ (t, s)(E[|G|[2n,2m]| (2n, 2m; z)|3s ])1/3 ,

(7.34)

for all z ∈ C with 0 < ||z| − 1| < 1/2. Proof For definiteness, assume that m  n + 2, the case n  m + 2 being [x,y] similar. Using the definition of U ω (7.13), we see that U ω = U ω(−∞,2n−1] ⊕ U ω[2n,∞) + ne ,

(7.35)

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415

where ne is given by

⎧ (rt − t)e−iθ2n−2 , ⎪ ⎪ ⎪ ⎪ 2 −iθ2n−2 ⎪ , ⎪ ⎪−t e ⎪ ⎪ 2 −iθ2n−1 ⎪ (r − r)e , ⎪ ⎪ ⎪ −iθ ⎪ 2n−1 ⎪ , ⎨−rte ne (k, l) = rte−iθ2n , ⎪ ⎪ ⎪ (r2 − r)e−iθ2n , ⎪ ⎪ ⎪ ⎪ ⎪ −t2 e−iθ2n+1 , ⎪ ⎪ ⎪ ⎪ ⎪ (−rt + t)e−iθ2n+1 , ⎪ ⎪ ⎩ 0,

k = 2n − 2, l = 2n − 1 k = 2n − 2, l = 2n k = 2n − 1, l = 2n − 1 k = 2n − 1, l = 2n k = 2n, l = 2n − 1 k = 2n, l = 2n k = 2n + 1, l = 2n − 1 k = 2n + 1, l = 2n otherwise.

(7.36)

Denote Gn (z) = G(−∞,2n−1] (z) ⊕ G[2n,∞) (z). By the first resolvent identity, we have G(z) − Gn (z) = −G(z)ne Gn (z).

(7.37)

Therefore, it follows for all m  n + 2 that G(2n, 2m; z) = {1 + t2 e−iθ2n−2 G(2n, 2n − 2; z) + rte−iθ2n−1 G(2n, 2n − 1; z) −(r2 − r)e−iθ2n G(2n, 2n; z) − (t − rt)e−iθ2n+1 G(2n, 2n + 1; z)} ×G[2n,∞) (2n, 2m; z).

(7.38)

A similar application of the first resolvent identity, this time to the difference G[2n,∞) (z) − G[2n,2m] (z) ⊕ G[2m+1,∞) (z), allows to express G[2n,∞) (2n, 2m; z) in terms of G[2n,2m] (2n, 2m; z) and ultimately leads to ) G(2n, 2m; z) = 1 + t2 e−iθ2n−2 G(2n, 2n − 2; z) + rte−iθ2n−1 G(2n, 2n − 1; z) * − (r2 − r)e−iθ2n G(2n, 2n; z) − (t − rt)e−iθ2n+1 G(2n, 2n + 1; z) ) × 1 − e−iθ2m [(rt − t)G[2n,∞) (2m − 1, 2m; z) + (r2 − r)G[2n,∞) (2m, 2m; z) + rtG[2n,∞) (2m + 1, 2m; z) * − t2 G[2n,∞) (2m + 2, 2m; z)] G[2n,2m] (2n, 2m; z). (7.39) If A and B denote the two {·}-factors on the right hand side of (7.39), then it follows from s < 1/2 and Theorem 3.1 that E(|A|3s )  C,

E(|B|3s )  C

(7.40)

uniformly in |z| = 1. Here we are using that Theorem 3.1 remains true with identical proof for the Green function of U ω[2n,∞) . An application of Hölder’s inequality to (7.39) yields 1/3  E[|G(2n, 2m; z)|s ]  (E[|A|3s ])1/3 (E[|B|3s ])1/3 E[|G[2n,2m] (2n, 2m; z)|3s ] , (7.41)

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which gives (7.34) when combined with (7.40).

Step Three Now that we have reduced the problem to dealing with fractional |[2n,2m]| moments of elements of the form Gz (2n, 2m), we show that expectations of such objects decay exponentially, in particular we show; Lemma 7.5 Assume that {θkω }k∈Z are i.i.d. with probability measure dμ(θ ) = τ (θ)dθ, where τ ∈ L∞ (T). There exist s0 ∈ (0, 1), 0 < C1 < ∞, α1 > 0 such that E[|G|[2n,2m]| (2n, 2m; z)|s0 ]  C1 e−α1 |m−n| ,

(7.42)

for all z ∈ C with 0 < ||z| − 1| < 1/2, and all m, n ∈ Z such that |m − n|  2. Proof For m  n + 2, let ϕ 2n and ϕ 2m be two solutions that satisfy the bound2n 2m = 1 and ϕ2m = 1. Using ary conditions at 2n and 2m, respectively, such that ϕ2n (7.18), we have G[2n,2m] (2n, 2m; z) =

eiθ2m 4 ϕ 2m . 2n 2m 2n 2m 4 ϕ2m+1 4 ϕ2m −4 ϕ2m 4 ϕ2m+1 2m

(7.43)

Since ϕ 2m satisfies the boundary condition at 2m, it follows that ω

2m 4 ϕ2m = tzeiθ2m ,

(7.44) ω

2m 4 ϕ2m+1 = (r − 1)zeiθ2m .

(7.45)

Using this along with the definition of 4 ϕ 2n , we obtain G[2n,2m] (2n, 2m; z) =

eiθ2m . ω 2n 2n tϕ2m−1 + (r − zeiθ2m )ϕ2m

(7.46)

Now, for s ∈ (0, 1) the expectation of the s-moment of G[2n,2m] (2n, 2m; z) is given by . / 2π 1 , E[|G[2n,2m] (2n, 2m; z)|s ] =  E dμ(θ2m ) 2n ω 2n s |tϕ2m−1 + (r − zeiθ2m )ϕ2m | 0 (7.47) where  E is the expectation with respect to the random variables {θkω }k∈Z\{2m} . By Lemma 7.2, we have ,6  2n  6−s 6 6 ϕ [2n,2m] s 6 6 . E(|Gz (2n, 2m)| )  Cμ (s, t)E 6Tz (ω, m − n) 2n−1 (7.48) 2n ϕ2n 6 Using Lemma 7.1, it follows that there exist α1 > 0 and s0 ∈ (0, 1) and 4(1) (s0 , t) < ∞ such that C μ 4(1) (s0 , t)e−α1 (m−n) , E[|G[2n,2m] (2n, 2m)|s0 ]  C z μ

(7.49)

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for all z ∈ C with 0 < ||z| − 1| < 1/2, 0 <  < 1/2 and m, n ∈ Z such that m − n  2. In the case n  m + 2 let ψ 2n and ψ 2m be solutions of (U − z)ψ = 0 that 2n 2m satisfy the boundary condition at 2n and 2m, respectively, with ψ2n = ψ2m = 1. Using (7.18), (7.14), we obtain that

G[2m,2n] (2n, 2m; z) =

2n z(tψ2m−1

1 . 2n + (1 − r)ψ2m )

(7.50)

ω In order to bound the expectation we first integrate with respect to θ2m−1 and use the same procedure as before.  

Proof of Theorem 3.4 Without restriction we may assume |k − l| > 4 (as (3.15) for |k − l|  4 only requires the a-priori bound (3.5)). Pick m, n ∈ Z such that k ∈ {2n, 2n + 1}, l ∈ {2m, 2m + 1} and |m − n| > 1. Thus using the results of Lemma 7.3 and Lemma 7.4, there exists 0 < κ(t, s, μ) < ∞ such that

E[|G(k, l; z)|s ]  κ(t, s, μ)

1  

1/12

E[|G|[2n+2i,2m−2 j]| (2n + 2i, 2m − 2 j; z)|12s ]

.

i, j=0

(7.51) 41 < ∞, Next the result of Lemma 7.5 gives that there exist s0 ∈ (0, 1), 0 < C α > 0 such that

4(1) κ(t, s0 , μ) E[|G(k, l; z)|s0 ]  C

1 

e−α|2n−2m+2i+2 j| .

(7.52)

i, j=0

Using the definition of m, n this yields (3.15).

 

8 Neumann Boundary Conditions In this section we study in detail the properties of one of the finite volume restrictions of the “free” unitary operator S introduced in Section 7.1. This particular restricted operator will share many of the properties of selfadjoint Neumann Laplacians. We will therefore think of them as restrictions of S to finite volume with Neumann boundary conditions.

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8.1 Neumann Boundary Conditions for d = 1 Let eiη = r + it, L ∈ N, and on l 2 ([0, 2L − 1]) define ⎛ iη rt −t2 re ⎜ −teiη r2 −rt ⎜ ⎜ rt r2 rt −t2 ⎜ 2 ⎜ −t −rt r2 −rt ⎜ ⎜ [0,2L−1] .. SN =⎜ . ⎜ ⎜ rt r2 ⎜ 2 ⎜ −t −rt ⎜ ⎝

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ 2 ⎟ rt −t ⎟ ⎟ r2 −rt ⎟ rt r2 teiη ⎠ −t2 −rt reiη (8.1) which is a special case of the restrictions of S0 to finite intervals defined in define λk = arccos(r2 − Section 7.1. To characterize the spectrum of S[0,2L−1] N 2 t cos(kπ/L)), k = 0, 1, . . . , L, i.e. in particular λ L = 0 and λ0 = arccos(r2 − t2 ), the latter coinciding with λ0 as introduced in Section 2 and giving the band edges of S0 as e±iλ0 . Proposition 8.1 The 2L eigenvalues of S[0,2L−1] are non degenerate and N given by σ (S[0,2L−1] ) = {eiλ0 , eiλL = 1} ∪ {e±iλk : k = 1, .., L − 1}. N

(8.2)

) ⊂ σ (S). Moreover, In particular, σ (S[0,2L−1] N ϕ0 = (i, 1, i, 1, · · · , i, 1)t

(8.3)

is an eigenvector associated with eiλ0 and ϕ L = (i, 1, −i, −1, · · · , (−1) L+1 i, (−1) L+1 )t

(8.4)

is an eigenvector associated with 1. To prove Proposition 8.1 we will use the transfer matrix formalism, i.e. solutions of U ω ψ = zψ are characterized by the relations      ω ω  ψ2k−1 ψ2k+1 = Tz θ2k , θ2k+1 , (8.5) ψ2k+2 ψ2k for all k ∈ Z, where the transfer matrices Tz : T2 → GL(2, C) are defined by (7.3). In the “free” case S0 ψ = eiλ ψ, the transfer matrix takes the simple form ⎛ ⎞  r −e−iλ 1 − e−iλ ⎜ ⎟ t T(λ) := Teiλ (0, 0) = ⎝ r  (8.6) iλ 2  ⎠.   e r 1 − e−iλ − 2 + 2 2 − e−iλ t t t The following lemma will be used in the proof of Proposition 8.1.

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Lemma 8.1 The vector (i, 1)t is an eigenvector of T(λ) L if and only if λ ∈ {0, arccos(r2 − t2 )} ∪ {λ : cos(λ) = r2 − t2 cos(kπ/L), k = 1, .., L − 1}. Proof In order to simplify the analysis we distinguish between two cases: (i) (i, 1)t is an eigenvector of T(λ). A straightforward calculation shows that this is only true for λ ∈ {0, arccos(r2 − t2 )} with corresponding eigenvalues {−1, 1} respectively. (ii) The second case is when (i, 1)t is an eigenvector of T(λ) L but not of T(λ), i.e.     i i =a , (8.7) T(λ) L 1 1 while v = T(λ)(i, 1)t is linearly independent of (i, 1)t . Then it follows that T(λ) v = T(λ) L

L+1

    i i = aT(λ) = av. 1 1

(8.8)

Also since v and (i, 1)t are linearly independent it follows that T(λ) L = aI. Finally using that the determinant of T(λ) is one, we see that a2 = 1. 2 2 Since the eigenvalues of T(λ) are given by e±i arccos((r −cos(λ))/t ) , we have that 2 2 a is an eigenvalue of T(λ) L if and only if e±i2L arccos((r −cos(λ))t ) = 1, i.e. λ ∈ {cos(λ) = r2 − t2 cos(kπ/L), k = 1, .., 2L − 1} = {cos(λ) = r2 − t2 cos(kπ/L), k = 1, .., L}. Combining the two cases gives the result.

(8.9)  

Proof of Proposition 8.1 In light of the previous Lemma, it suffices to show that eiλ ∈ σ (S[0,2L−1] ) if and only if (i, 1)t is an eigenvector of T(λ) L . N First it is not hard to see that eiλ ∈ σ (S[0,2L−1] ) means the existence of ψ ∈ N l 2 ([0, 2L − 1]) such that     ψ2m+1 ψ2m−1 = T(λ) , (8.10) ψ2m+2 ψ2m for m ∈ {1, .., L − 2}, while 

ψ1 ψ2



and 

ψ2L−3 ψ2L−2



⎛

⎞ 1 i(η−λ) (r − e ) ⎜ ⎟ t = ψ0 ⎝ 1 ⎠, 2 i(η−λ) iλ iη (r − re − e + re ) t2

(8.11)

⎞ 1 2 i(η−λ) iλ iη (r − re − e + re ) ⎟ ⎜ 2 = ψ2L−1 ⎝ t ⎠. −1 i(η−λ) ) (r − e t

(8.12)

⎛

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Define ψ˜ −1 such that T(λ)−1



ψ1 ψ2



 =

ψ˜ −1 ψ0

 .

(8.13)

Using (8.11) and that eiη = r + it, one can see that this definition is equivalent to having ψ˜ −1 = iψ0 . Similarly, defining ψ˜ 2L such that     ψ2L−1 ψ2L−3 = T(λ) , (8.14) ψ2L−2 ψ˜ 2L we see that ψ˜ 2L = −iψ2L−1 . Also by definition,     ψ2L−1 ψ˜ −1 T(λ) L = ψ0 ψ˜ 2L     −iψ2L−1 i i = . T(λ) L 1 1 ψ0

(8.15) (8.16)

which shows the required assertion. The last two claims of Proposition 8.1 follow from the above together with T(0)(1, i)t = −(1, i)t and T(λ0 )(1, i)t =   (1, i)t . We conclude this subsection with several remarks: (i) ei arccos(r −t ) ∈ σ (S[0,2L−1] ), while e−i arccos(r −t ) is not. N 2 2 ϕ0 = eiλ0 ϕ0 has solu(ii) For λ0 = arccos(r − t ) = arccos(1 − 2t2 ), S[0,2L−1] N tions with |ϕ0 (k)| = 1 for all k ∈ [0, 2L − 1]. , given (iii) There is a gap between the upper edge of the spectrum of S[0,2L−1] N 2 2 by ei arccos(r −t ) , and the next closest eigenvalue. In particular, for any k ∈ {1, · · · , L − 1} we have 2

2

|ei arccos(r

2

2

−t2 )

− ei arccos(r

2

−t2 cos(π k/L))

2

| > t2 (1 − cos(π k/L)) = 2t2 sin2 (π k/2L)  2 (4 − π )2 2 πk t , L 32

(8.17)

using the property sin(x)  x(1 − π/4) if x ∈ (0, π/2). (iv) The previous remarks as well as the “Neumann-bracketing” property to a suitable be found in Section 10 below will make the operators S[0,2L−1] N tool when studying properties of finite volume restrictions of the unitary Anderson model near the upper edge eiλ0 of the spectrum of S0 . To get the corresponding results also at the lower edge e−iλ0 of the spectrum of S0 one needs to modify the definition of S[0,2L−1] by setting eiη = r − it N in (8.1). In this case we use, in particular, that the vector (−i, 1)t is an eigenvector of T(−λ0 ), leading to e−iλ0 becoming an eigenvalue of the restricted operator. One gets properties similar to Proposition 8.1 and Remarks (i), (ii) and (iii) above.

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8.2 Neumann Boundary Conditions for d > 1 For a box  := [2l1 , 2m1 − 1] × . . . × [2ld , 2md − 1] ⊂ Zd define [2l ,2m j −1]

j d S N = ⊗ j=1 S N

on

⊗dj=1 l 2 ([l j, 2m j − 1]) = l 2 ().

(8.18)

Note here that the discussion of Section 8.1 applies with obvious modifications to intervals of the form [2l j, 2m j − 1] and integers l j < m j. We will be particularly interested in the case of cubic boxes  L := [−2L, 2L + 1]d for L ∈ N. L d The spectrum of S N is given by the | L | = (4L + 2) eigenvalues 8 ) * 7 iσ j λk j d L =  (8.19) σ S e j=1 N where k j ∈ {0, 1, 2, · · · , 2L + 1}d for j = 1, . . . , d, σ j ∈ {+1, −1} for k j = 1, . . . , 2L,

σ j = 1 for k j ∈ {0, 2L + 1}.

(8.20)

Under the assumption d arccos(r2 − t2 ) < π , the upper edge of the spec2 2 trum of S, eid arccos(r −t ) = eidλ0 , belongs to σ (S N ) and is non degenerate. An eigenvector corresponding to eidλ0 is ϕ0(d) = ⊗d1 ϕ0 , whose components all have modulus one. Moreover, there exists c0 , a numerical constant, such that 7 8

7 8 c0 t2 L . (8.21) \ eidλ0 > dist eidλ0 , σ S N | L |2/d Indeed, the closest eigenvalue to eidλ0 is ei((d−1)λ0 +arccos(r is d-fold degenerate, so that the distance (8.21) equals |eidλ0 − ei((d−1)λ0 +arccos(r

2

−t2 cos(π/(2L+1))))

2

−t2 cos(π/(2L+1))))

| = |eiλ0 − ei arccos(r >

2

2

, which

−t2 cos(π/(2L+1)))

|

2

t c0 t c0 = , (4L + 2)2 | L |2/d

(8.22)

where c0 = (π(4 − π ))2 /8, see (8.17). For later study of spectral properties near the lower edge e−idλ0 of the [2l ,2m −1] spectrum of S, we use the modified version of S N j j from the fourth remark at the end of the previous subsection in the definition (8.18).

9 The Feynman-Hellmann Formula The Feynmann-Hellmann formula provides, on the level of first order perturbation theory, the change of an isolated simple eigenvalue of a selfadjoint operator under an additive perturbation. Here we will need a corresponding result for multiplicative perturbations of unitary operators. We prove such a formula in an analytic framework, which will suffice for our purpose.

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Proposition 9.1 Let I ⊂ R be an open interval containing zero, H be a separable Hilbert space and I  α → U(α) an analytic map with values in the set of unitary operators on H. Assume β(0) ∈ S is an isolated simple eigenvalue of U(0) with normalized corresponding eigenvector ϕ(0) ∈ H. Then, there exists an open disc centered at 0 of radius α0 > 0, D(0, α0 ) ⊂ C, and two analytic maps D(0, α0 )  α → β(α) ∈ C and D(0, α0 )  α → ϕ(α) ∈ H such that U(α)ϕ(α) = β(α)ϕ(α) ∀α ∈ D(0, α0 ), dist (β(α), σ (U(α) \ {β(α)}) > 0

ϕ(α) = 1 if α ∈ D(0, α0 ) ∩ I.

(9.1)

Moreover, for all α ∈ D(0, α0 ) ∩ I, β  (α) = ϕ(α)|U  (α)ϕ(α).

(9.2)

Remark 9.1 i) For a given α, the last formula is of course true for any choice of normalized eigenvector of U(α), corresponding to β(α). ii) If I  α → U(α) is analytic and takes its values in the set of unitary finite matrices, all its eigenvalues and spectral projectors admit analytic extensions in a complex neighborhood of I, even at the values of α where eigenvalues of U(α) may cross, see [42]. Consequently, an analytic choice of normalized eigenvectors can be made for all α ∈ I. Proof By the general theory of analytic perturbations of operators, see e.g. [42], the operator U(α) admits an isolated simple eigenvalue β(α), for small enough values of |α|, say in D(0, α0 ). Also, the analytic rank one spectral projector on β(α), P(α), given by the Riesz formula is analytic for all α ∈ D(0, α0 ). By definition, for all α ∈ D(0, α0 ), P(α)U(α) = U(α)P(α) = β(α)P(α),

(9.3)

and since U(α) is unitary on the real axis, P(α) is self-adjoint for real α’s. Now define the analytic operator W(α) as the unique solution to the ODE W  (α) = [P (α), P(α)]W(α),

W(0) = I,

α ∈ D(0, α0 ).

(9.4)

It is a well known property ([42]) that the following intertwining property holds for all α ∈ D(0, α0 ), P(α)W(α) = W(α)P(0).

(9.5)

Note that W(α) is unitary on the real axis, since its generator is easily seen to be anti self-adjoint there. We define an analytic vector by ϕ(α) = W(α)ϕ(0).

(9.6)

Identities (9.5) and (9.3) show that U(α)ϕ(α) = β(α)ϕ(α)

(9.7)

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423

and ϕ is normalized on the real axis, since W is unitary there. By differentiation of the previous identity and application of P(α) to the result, we obtain P(α)U  (α)ϕ(α) + P(α)U(α)ϕ  (α) = β  (α)P(α)ϕ(α) + β(α)P(α)ϕ  (α)

(9.8)

which reduces to P(α)U  (α)ϕ(α) = β  (α)ϕ(α)

(9.9)

due to (9.3) and (9.5). Since for all α ∈ I ∩ D(0, α0 ) we can write P(α) = |ϕ(α)ϕ(α)| = W(α)|ϕ(0)ϕ(0)|W −1 (α),

(9.10)  

the result follows.

As a specific application, let us consider the family of analytic unitary matrices −iαθk U  (α) = D(α)S }S N = diag{e N,

(9.11)

where S N is the Neumann restriction of S to a d-dimensional box  introduced in Section 8, α ∈ R, and θk ∈ T for all k ∈ . U  (α) interpolates between S N  and diag{e−iθk }S N , at α = 0 and α = 1, respectively. Introducing the self-adjoint  matrix H  = k∈ θk |ek ek | on l 2 (), we can rewrite 

U  (α) = e−iα H S N , α ∈ R.

(9.12)

Lemma 9.1 If eiλ(0) is a discrete non-degenerate eigenvalue of S N with normalized eigenstate ϕ(0), then, for all α ∈ R, there exist analytic eigenvalues eiλ(α) of U  (α) with analytic normalized eigenvectors ϕ(α) such that  d iλ(α) e = −ieiλ(α) θk |ek |ϕ(α)|2 . dα

(9.13)

k∈

In particular, for all α ∈ R, λ (α) = −



k∈ θk

|ek |ϕ(α)|2 .

Proof The existence of analytic eigenvalues eiλ(α) and analytic eigenvectors ϕ(α) of U  (α), α ∈ R, follows from Proposition 9.1 and the remark following it. We compute (U  ) (α) = −iH U  (α), and ϕ(α)| − iH  U  (α)ϕ(α) = −i

U  (0) = S N 

θk |ek |ϕ(α)|2 eiλ(α)

(9.14)

(9.15)

k∈

and apply (9.2).

 

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10 Splitting Boxes by Neumann Boundary Conditions Throughout this section we will assume that boxes  ⊂ Zd are compatible with Neumann boundary conditions as defined in Section 8, S N is given by (8.18) and −iθk U  = DS )S N = diag(e N.

(10.1)

For notational simplicity we will assume in this section that the box  has a vertex at the origin, which does not cause a restriction. We first deal with dimension d = 1. Consider a one dimensional box 0 consisting of two disjoint adjacent boxes 1 and 2 : 1 = [0, 2l − 1], 2 = [2l, 2(l + n) − 1], 0 = [0, 2(l + n) − 1], S[0,1] N ).

0

(10.2) 1

We note that U and U ⊕ U 2 with n, l  2 (to avoid the special case 2 are both defined on l ([0, 2(l + n) − 1]). We want to show that the eigenvalues of U 1 ⊕ U 2 are closer to the upper band edge ei(λ0 +a) of the almost sure spectrum of U than those of U 0 . Recall here that in Section 3.2.3 we have assumed that |θk |  a and λ0 + a < π . This is the analog of the well known property H 1 ⊕ H 2  H 0 , where H  is the Neumann restriction to a box  of the discrete Schrödinger operator. The following simple observation is the starting point of the analysis. Splitting a box by imposing Neumann boundary conditions is a rank one perturbation: 

Lemma 10.1 Let S N j , j = 0, 1, 2 be defined as above. Then, 1 2 0 S N = S N ⊕ S N + |ψϕ|

(10.3)

where ψ = −te2l−2 − re2l−1 − ire2l + ite2l+1 ϕ = t(−ie2l−1 + e2l )

(10.4)

The proof is an easy computation. This leads us to using the following fact about rank one perturbations of unitary operators which return a unitary operator. Lemma 10.2 Let U a unitary operator on a Hilbert space H and f, g ∈ H \ {0}. If V = U + | f g|

(10.5)

is unitary, then there exists β ∈ (−π, π ] such that eiβ = 1 + Ug| f  and ˆ

ˆ

V = eiβ| f  f | U, where fˆ = f/ f .

(10.6)

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Proof The identity VV ∗ = I implies that |Ug f | + | f Ug| + g 2 | f  f | = 0.

(10.7)

Applying this to f shows that Ug is proportional to f , so that Ug =

 f |Ug f.

f 2

(10.8)

With this it follows from (10.5) that V = (I + | f Ug|)U = (I + Ug, f | fˆ fˆ|)U = (I − | fˆ fˆ| + μ| fˆ fˆ|)U,

(10.9)

where μ = 1 + Ug| f . Thus I − | fˆ fˆ| + μ| fˆ fˆ| = VU ∗ is unitary, which shows that |μ| = 1, i.e. μ = eiβ for β ∈ (−π, π ], and I − | fˆ fˆ| + μ| fˆ fˆ| = ˆ ˆ eiβ| f  f | .   Taking into account the random phases, we apply the previous lemma to our case with U 0 = U 1 ⊕ U 2 + |Dψϕ|

(10.10) √ and√ψ, ϕ from (10.4). We compute Dψ = ψ = 2, i.e. ψˆ = ψ/ ψ = ψ/ 2, and 2 −i arccos(r 1 eiβ = 1 + (U 1 ⊕ U 2 )ϕ|Dψ = 1 + (S N ⊕ S N )ϕ|ψ = e

2

−t2 )

= e−iλ0 (10.11)

so that ˆ

ˆ

ˆ

ˆ

2 1 U 0 = e−iλ0 |DψDψ| U 1 ⊕ U 2 = De−iλ0 |ψψ| S N ⊕ SN .

(10.12)

Let us introduce an analytic family of unitary operators defined in I, a complex neighborhood of [0, 1], by ˆ

ˆ

I  α → U(α) = e−iαλ0 |DψDψ| U 1 ⊕ U 2 ,

(10.13)

such that U(0) = U 1 ⊕ U 2 and U(1) = U 0 . By Lemma 9.1 we immediately get the following Proposition 10.1 Let α ∈ I and eiλ(α) denote any analytic eigenvalue of U(α), which is isolated except at a finite set of values of α. Then arg(λ(1))  arg(λ(0)).

(10.14)

Remark 10.1 In other words, when a Neumann boundary condition is introduced to split 0 into 1 ∪ 2 , the eigenvalues of U 1 ⊕ U 2 are closer to ei(λ0 +a) than those of U 0 .

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Let us generalize now to dimension d  1. Consider a box 0 of the form 0 = 0 (1) × (2) × . . . (d)

(10.15)

where 0 (1) = [0, 2(l + n) − 1], ( j) = [0, 2l j − 1], j = 2, · · · , d,

(10.16)

which we split by a Neumann boundary condition perpendicular to the first axis as 0 = 1 ∪ 2 with k = k (1) × (2) × . . . (d), k = 1, 2

(10.17)

and 0 (1) = 1 (1) ∪ 2 (1) = [0, 2l − 1] ∪ [2l, 2(l + n) − 1]. By the previous results, the corresponding operators U related by ˆ

k

(10.18)

, k = 0, 1, 2 are

ˆ

U 0 = e−iλ0 |DψDψ|⊗I⊗···⊗I U 1 ⊕ U 2 .

(10.19)

Applying Lemma 9.1 again, with H replaced by the non negative operator ˆ ˆ ⊗ I ⊗ · · · ⊗ I, shows that the spectra of U 0 and U 1 ⊕ U 2 are |DψD ψ| related in the same way as in the one dimensional case, e.g. by (10.14). Clearly, the splitting by Neumann boundary conditions can be done perpendicular to any of the d coordinate axes and can also be iterated. Thus we get the above form of spectral monotonicity also, for example, when spitting U L over the cube [−2L, 2L + 1]d into a direct sum of U i for ((2L + 1)/l)d cubes i of sidelength 2l.

11 A Combes-Thomas Estimate Combes-Thomas bounds, originating from [19], have become the standard tool in Schrödinger operator theory to show exponential decay of eigenfunctions to eigenvalues which lie outside of the essential spectrum. They also provide a key step in localization proofs for random Schrödinger operators in the band edge regime, see e.g. [56]. Here we provide a Combes-Thomas type estimate for unitary operators with band structure. Let U be unitary on l 2 (Zd ). We say that U has band structure of width w > 0, if it can be written as U = D + O with ek |De j = δkjek |Dek  and ek |Oe j = 0 if | j − k| > w. (11.1) Proposition 11.1 (Combes-Thomas type estimate) For a unitary operator U on l 2 (Zd ) with band structure of width w, there exist 0 < B < ∞ which depends on U only, such that |e j|(U − z)−1 ek | 

2 e−dist(z,σ (U))| j−k|B . dist(z, σ (U))

(11.2)

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Remark 11.1 i) The same result holds for U defined on a finite dimensional Hilbert space l 2 (),  ⊂ Zd , with constants independent of . ii) Actually, our proof works more generally for bounded normal operators U with band structure. Results of this type are known in the literature, e.g. [25]. Proof Let x = (x1 , · · · , xd ), where xn is the self-adjoint multiplication operator acting on ek , with k = (k1 , . . . , kd ), as xn ek = kn ek and defined on its natural domain. We introduce the vector α = (α1 , · · · , αd ) ∈ Rd and construct the selfadjoint operator eαx acting as eαx ek = eαk ek on Dα = {ψ ∈ l 2 (Zd ) s.t.

Here αk =



|ek |ψ|2 eαk < ∞}.

(11.3)

(11.4)

k∈Zd

 n

αn kn . Consider the operator

U α := eαx Ue−αx = eαx De−αx + eαx Oe−αx = Dα + Oα

(11.5)

defined a priori on the dense set c0 = {ψ ∈ l 2 (Zd ) s.t. ek |ψ = 0 for |k| large enough}. The operator U α is bounded because for any ψ ∈ c0   Oα ψ = eα(k− j) ek |Oe je j|ψek j∈F

(11.6)

(11.7)

k= j |k− j|w

where the set F is finite and eα(k− j)  e|α|w . Moreover, Dα = D which shows that U α  C1 (α)  C1 < ∞ on c0 , for α in a bounded set. Similarly,

U − U α = O − Oα  C2 (α)  C3 |α|,

(11.8)

for |α| small enough. From the resolvent identity, if z ∈ ρ(U) and C3 |α|/dist (z, σ (U)) < 1/2, (U α − z)−1 = (U − z)−1 (I + (U α − U)(U − z)−1 )−1

(11.9)

(U α − z)−1  2 (U − z)−1  2/dist (z, σ (U)).

(11.10)

with

Finally, by the formula (U α − z)−1 = eαx (U − z)−1 e−αx ,

(11.11)

we derive  j|eαx (U − z)−1 e−αx k = e−α(k− j)  j|(U − z)−1 k =  j|(U α − z)−1 k,

(11.12)

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from which we get | j|(U − z)−1 k|  2

e+α(k− j) . dist (z, σ (U))

(11.13)

Choosing the components of α and their sign in such a way that |αn | = |α| > 0 and α( j − k) =

d 

αn ( j − k)n  |α|| j − k|,

(11.14)

n=1

we obtain the result, with |α| = dist (z, σ (U))/4C3 , and B = 1/4C3 .

 

12 The Genesis of Lifshits Tails After having introduced some tools in the previous two sections we will now start with the actual proof of Theorem 3.5. Throughout this proof we will focus on localization at the upper band edge ei(dλ0 +a) of the almost sure spectrum of U ω . The proof at the lower band edge is completely analogous. It uses the alternate form of Neumann boundary conditions discussed in Section 8 (setting eiη = r − it in (8.1) rather than r + it) and a corresponding adjustment of the results on splitting boxes in Section 10. We find it convenient to rotate the upper band edge of to be identical with eidλ0 , the upper band edge of S. This is achieved by replacing the original U ω by e−ia U ω . In other words, setting θ M = 2a we now assume supp μ ⊂ [0, θ M ] with 0 ∈ supp μ and

2dλ0 + θ M < 2π.

(12.1)

The latter means that has the gap {eiϑ : dλ0 < ϑ < 2π − (dλ0 + θ M )}. As in earlier sections we will frequently drop the subscript ω from our notation. We will first establish a Lifshits tail estimate for the spectrum near the band edge eidλ0 . At the root of this is the following proposition which we prove by following the steps of Stollmann [56]. As in Section 8, for L ∈ N we set  L = [−2L, 2L + 1]d . 

Proposition 12.1 Let eiλ(U L ) , respectively eidλ0 , be the eigenvalue of largest L L argument of U L , respectively S )  dλ0 and there exist b > 0 N . Then λ(U and γ > 0, independent of L and d, such that   b L d P |eiλ(U ) − eidλ0 |  2  e−γ L , (12.2) L for L large enough. Let us first give an easy Corollary of Lemma 9.1. Recall that U L (α) is defined by (9.11).

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Lemma 12.1 Consider a fixed realization of U L (α) in dimension d  1. Then the analytic continuation of any eigenvalue eiλ(α) of U L (α) is such that λ(α) is non increasing. Consequently, eiλ(α) − eiλ(0) is a non decreasing function of α  0, as long as λ(0) − λ(α) < π . L iλ0 (α) Moreover, the eigenvalue eidλ0 of S N and its analytic continuation e satisfy  d 1 λ0 (α)|α=0 = − θk . (12.3) d dα (4L + 2) k∈ L

Proof The first statement follows from Lemma 9.1 and from d iλ(α) 2 − eiλ(0) = 2 sin(λ(α) − λ(0))λ (α). (12.4) e dα The second statement makes use of the fact that the components of the   eigenvector ϕ0(d) all have equal modulus. Recall the following standard large deviation estimate whose proof can be found, e.g., in Lemma 2.1.1 of [56]. Lemma 12.2 For non-trivial and non-negative i.i.d. random variables θk and s0 = γ0 = − 21 ln(E(e−θ0 )) > 0, we have 9 : 1  P θi  s0  e−γ0 || . (12.5) || i∈ Let us consider the small α behavior of eiλ0 (α) . Lemma 12.3 There exist c1 > 0 and c2 > 0, independent of d and L, such that for L sufficiently large   d iλ0 (α) c2 iλ0 (α) idλ0 −e −α e  c1 α 2 L2 , 0  α  2. (12.6) e dα L α=0 Proof Expanding eidλ0 (α) in terms of α ∈ R, we get that d iλ0 (0) α 2 d2 iλ0 (α) e = e ˜ dα 2 d2 α for some 0 < α˜ < α. Next we use Cauchy’s integral formula to bound   eiλ0 (z) eiλ0 (z) − eidλ0 d2 iλ0 (α) 2! 2! ˜ e = dz = dz d2 α 2πi |z−α|=r (z − α) ˜ 3 2πi |z−α|=r (z − α) ˜ 3 ˜ ˜ eiλ0 (α) − eidλ0 − α

(12.7)

(12.8)

for α˜ small enough and suitable r > 0. Thus we need to control eiλ0 (z) for z complex. Since U L (α) − U L (0) = (e−iα H

L

− I)S N,

(12.9)

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where

e−iα H

L

− I  e|α|θ M − 1,

(12.10)

we have, by the second resolvent identity, −1 −iα H L (U L (α) − z)−1 = (S N − z) (I − (e

L

L − I)S (α) − z)−1 ) (12.11) N (U

L for z ∈ σ (U L (α)) ∪ σ (S N ). Hence,

|α|θ M L dist (z, σ (S − 1 =⇒ z ∈ ρ(U L (α)). N )) > e

(12.12)

Now (8.21) says idλ0 L }) > δ = dist (eidλ0 , σ (S N ) \ {e

Thus, if |α| < α0 :=

t2 c0 . | L |2/d

(12.13)

ln(1+δ/2) , θM

{z | |z − eidλ0 | = δ/2} ⊂ ρ(U L (α)).

(12.14)

We now take α ∈ (−α0 /2, α0 /2) so that α˜ < α0 /2 and r = α0 /2 so that {z | |z − α| ˜ = α0 /2} ⊂ {z | |z|  α0 }

(12.15)

and for such z’s |eiλ0 (z) − eidλ0 | < δ/2. Using δ/4 < ln(1 + δ/2) < δ/2 if δ < 1/2, one gets that if 0  α < 8θδM ,  2 2 θM 2 α 2 d2 iλ0 (α) ˜ 2δ 2 e  α  α 32 . (12.16) 2 d2 α 2 α0 δ We get the announced result with c1 =

2 128θ M t2 c0 , c2 = , 2 t c0 32θ M

provided | L |2/d = (4L + 2)2 > 2t2 c0 , i.e. for L sufficiently large.

(12.17)  



Proof of Proposition 12.1 Assume that |eiλ(U L ) − eidλ0 |  b /L2 , with b to be determined later. Using the monotony in α (Lemma 12.1) and Lemma 12.3, we have for 0  α  c2 /L2 and L large enough   d iλ0 (α) α e  eiλ0 (α) − eidλ0 + c1 α 2 L2 dα α=0  eiλ(U 



)

− eidλ0 + c1 α 2 L2

b + c1 α 2 L2 . L2

(12.18)

Dividing by α and then choosing α = c4 /L2 such that c4  c2 and c1 c4  s0 /2, we obtain that   d iλ0 (α) e  b /c4 + s0 /2. (12.19) dα α=0

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Next we choose b such that b /c4  s0 /2 to get that   d iλ0 (α) e  s0 . dα α=0

(12.20)

Note that b is thus independent of d and L. On the other hand we have from (12.3) that    1 d iλ0 (α) = e θk . (12.21) dα (4L + 2)d α=0 k∈ L

In probabilistic terms )

ω | |eiλ(U

L

)

− eidλ0 |  b /L2

*

⎫ ⎬  1 ⊂ ω| θ  s k 0 . ⎭ ⎩ (4L + 2)d ⎧ ⎨

(12.22)

k∈ L

Finally an application of Lemma 12.2 ends the proof with γ = 2d γ0 .

 

The Lifshits tail estimate of Proposition 12.1 and the properties of the Neumann boundary conditions, Proposition 10.1, allow to prove the following result, which is based on an equivalent result for Schrödinger operators provided in [56]. Proposition 12.2 Let β ∈ (0, 1). There exist finite positive constants γ¯ , C and a sequence of positive integers Lk with Lk → ∞ such that for any k and any z ∈ C β with 1 < |z| < 2 and dλ0 − 1/Lk  arg z  dλ0 ,  d(1−β/2) −γ¯ Lkdβ/2 β P dist (z, σ (U Lk ))  1/Lk  CLk e . (12.23) Proof of Proposition 12.1 Let β ∈ (0, 1) and b > 0 the constant found in Proposition 12.1. Fix a constant C > 1. We claim that for each sufficiently large k ∈ N there exists Lk ∈ N which is a multiple of k and such that b 2 b  β  2. 2 Ck k Lk To see this, note that (12.24) is equivalent to

Lk ∈ (2k2 /b )1/β , (2k2 /b )1/β C1/β .

(12.24)

(12.25)

As β < 1, for k sufficiently large, this interval has length larger than k, allowing for a choice of Lk as required. We now show that (12.24) holds for these Lk . Split the box  Lk = [−2Lk , 2Lk + 1]d into M = | Lk |/|k | = (Lk /k)d disjoint boxes as  Lk = ∪ M j=1 k ( j),

(12.26)

where k ( j) denotes the suitably translated box k + c( j). Consider now U N = U k (1) ⊕ U k (2) ⊕ · · · ⊕ U k (M)

(12.27)

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2 k ( j) on l 2 ( Lk ) = ⊕ M is provided with Neumann j=1 l (k ( j)), where each U boundary conditions. Proposition 10.1 shows that passing from U Lk to U N by introducing Neumann boundary conditions makes the eigenvalues come closer to the upper band edge, i.e.

λ(U Lk )  λ(U k ( j0 ) ) for some j0 ∈ {1, 2, · · · , M},

(12.28)

where eiλ(U) denotes the eigenvalue of largest argument of U. As a consequence, taking the stochastic independence of the (U k ( j) ) into account together with the relation (12.24) 7 8 7 8 L ( ( j0 )) β P |eiλ(U k ) − eidλ0 |  2/Lk  P |eiλ(U k ) − eidλ0 |  b /k2 for some j0 7 8 ( (1))  MP |eiλ(U k ) − eidλ0 |  b /k2 . (12.29) By applying Proposition 12.1 to the box k (1) we see that the latter is bounded by Ldk −γ kd d(1−β/2) −γ¯ Lkdβ/2 e  CLk e . d k β

(12.30) β

Finally, since dist(z, σ (U Lk ))  1/Lk for z such that |z − eidλ0 |  1/Lk implies  iλ(U Lk

|e

)

β

− eidλ0 |  2/Lk , we get the result.

 

13 Towards an Iterative Proof of Exponential Decay The proof of Theorem 3.5, to be completed in Section 14, will proceed as follows: To prove exponential decay of E(|G(k, l; z)|s ) we will join the two sites k and l by a chain of boxes of side length L. For a suitable choice of L and arg z close to the edge of , the Lifshits tail and Combes-Thomas estimates will show that the fractional moment of the finite volume Green function G(L) (k, j; z) is small (think “less than one” even if this is only true up to some factors which can be controlled). Here G(L) is the resolvent of a restriction of U to a box of side length L centered at k and j is a boundary site of this box. To turn this into a proof of exponential decay of the infinite volume Green function, we need two more tools: (i) a factorization of the infinite volume Green function into finite volume factors, often referred to as a geometric resolvent identity, (ii) a decoupling argument which allows to factorize the fractional moments in the geometric resolvent identity. These two remaining tools will be provided in this section. As explained at the beginning of Section 12 we will continue to focus on the localization proof at the upper band edge and continue to assume (12.1), so that the upper edge of is eidλ0 .

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13.1 A Geometric Resolvent Identity Due to the specific structure of our operators (in particular their ergodicity with respect to translations by two) it is of advantage to cut up Zd into cubes of side length two. Thus, for n = (n1 , . . . , nd ) ∈ Zd let Cn := [2n1 , 2n1 + 1] × . . . × [2nd , 2nd + 1] and χn := χCn the characteristic function of Cn . For L ∈ N let ; L = Cn = [−2L, 2L + 1]d .

(13.1)

(13.2)

|n| L c

We will work with restrictions U ωL and U ω L of U ω to  L and its complement L c L  L = Zd \  L . We choose U ωL = Dω S N , where S N is the unitary Laplacian with Neumann boundary conditions from (8.18). In fact, the choice of boundary conditions is rather irrelevant as long as matrix elements are only affected near the boundary, e.g. we have U ωL ( j, k) = U ω ( j, k)

if j, k ∈  L−1 .

(13.3)

Our definition of Neumann operators from Section 8 does not directly extend to operators on exterior domains such as cL , where the unitary Laplacian can not be defined as a tensor product of one-dimensional Laplacians. While it is possible to define Neumann boundary conditions directly for the d-dimensional operator, we choose a more simplistic approach and define c

U ω L = PL U ω PL , c

c

(13.4)

cL

viewed as an operator on 2 (cL ). Here P denotes the orthogonal projection c onto 2 (cL ). The price for our simplemindedness is that U ω L is not unitary. c ( ) However, it is a contraction, i.e. U ω L  1 and therefore σ (U ω (cL )) ⊂ {z ∈ C : |z|  1}, and it remains a band matrix whose entries satisfy, by definition, c

U ω L ( j, k) = U ω ( j, k)

if j, k ∈ cL .

(13.5)

These properties will suffice for what we need in Section 14. We will use what is often referred to as a geometric resolvent identity, c relating the resolvents of U ω , U ωL and U ω L . Following an argument which for the selfadjoint Anderson model is used in [5], we start by defining the boundary operator Tω(L) through c

U ω = U ωL ⊕ U ω L + Tω(L) .

(13.6)

cL

By the above construction of U ωL and U ω , in particular (13.3) and (13.5), the operator Tω(L) has non-vanishing matrix-elements only near the boundary of  L , more specifically Tω(L) χx = χx Tω(L) = 0 if |x|  L − 1 or |x|  L + 2

(13.7)

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as well as χx Tω(L) χ y = 0

if |x − y|  2.

(13.8)

Also, the matrix-elements of Tω(L) are uniformly bounded in L and ω. To keep the length of the following equations under control we will drop the arguments ω and z and for the rest of this section write G := (U ω − z)−1

(13.9)

and c

c

G(L) := (U ωL ⊕ U ω L − z)−1 = (U ωL − z)−1 ⊕ (U ω L − z)−1 .

(13.10)

We do a double-decoupling, once on  L and once on  L+1 . Using the resolvent identity twice gives G = G(L) − G(L) T (L) G = G(L) − G(L) T (L) G(L+1) + G(L) T (L) GT (L+1) G(L+1) .

(13.11)

Observe that for y ∈ Zd with |y|  L + 2 one has χ0 G(L) χ y = 0 and χ0 G(L) T (L) G(L+1) χ y = 0. Thus χ0 Gχ y = χ0 G(L) T (L) GT (L+1) G(L+1) χ y ,

(13.12)

which is the geometric resolvent identity to be used below. 13.2 Decoupling of Fractional Moments The next result says that the fractional moment E( χ0 Gχ y s ) can be decoupled along the boundary of  L . Proposition 13.1 For every s ∈ (0, 1/3) there exists a constant C = C(s) < ∞ such that   E( χ0 Gχ y s )  C E( χ0 G(L) χu s ) E( χv G(L+1) χ y s ) (13.13) u: |u|=L

v  : |v  |=L+2

uniformly in z with 1 < |z| < 2, L ∈ N and y ∈ Zd with |y|  L + 2. Proof of Proposition 13.1 From here on, the symbol C will denote a generic constant which may change from line to line but which depends on inessential quantities only. Define the boundary of  L by ∂ L := {(x, y) ∈ Zd × Zd : χx T (L) χ y = 0} ⊂ {(x, y) ∈ Zd × Zd : L  |x|  L + 1, L  |y|  L + 1, |x − y|  1}. (13.14)

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Expanding (13.12) over the boundaries of  L and  L+1 gives 

χ0 Gχ y =

χ0 G(L) χu T (L) χu Gχv T (L+1) χv G(L+1) χ y .

(13.15)



(u, u ) ∈ ∂ L (v, v  ) ∈ ∂ L+1 Taking fractional moments and also using that T (L) and T (L+1) have uniformly bounded matrix-elements we get E( χ0 Gχ y s )  C



  E χ0 G(L) χu s χu Gχv s χv G(L+1) χ y s .

(u, u ) ∈ ∂ L (v, v  ) ∈ ∂ L+1 (13.16) Notice that the first and third factor in the expectation on the right are independent. Unfortunately, they are correlated via the middle factor. In order to decouple the factors we use a re-sampling argument, following a strategy developed in [3] and [14] as a tool in the fractional moments approach to continuum Anderson-type models. For this, fix two pairs (u, u ) ∈ ∂ L and (v, v  ) ∈ ∂ L+1 . Let J := Cu ∪ Cu ∪ Cv ∪ Cv . In the resolvents G(L) and G(L+1) we will re-sample the random variables θn , n ∈ J . For this choose i.i.d. random variables {θˆn }n∈J with the same distribution as the θn but independent from them.  Noting that Dω = n∈Zd e−iθn Pn (where Pn = en , ·en is the projection onto ˆ the canonical basis vector en ) we define the re-sampled Dω,ωˆ := Dω − D, where  ˆ ˆ := (e−iθn − e−iθn )Pn , (13.17) D n∈J

i.e. the variables {θn }n∈J are replaced by the corresponding θˆn . Also define (L) L L ˆ L U ω, ωˆ := Dω,ωˆ S N = U ω − DS N

(13.18)

L where U ω(L) = U ωL , U 0(L) = S N and

ˆ (L) := (U (L) − z)−1 . G ω,ωˆ

(13.19)

The resolvent identity yields ˆ (L) − G ˆ (L) DS ˆ L G(L) G(L) = G N

(13.20)

ˆ (L+1) . ˆ (L+1) − G(L+1) DS ˆ L+1 G G(L+1) = G N

(13.21)

and 

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We use this to bound the terms on the right of (13.16) by   E χ0 G(L) χu s χu Gχv s χv G(L+1) χ y s " ˆ (L) χu s + χ0 G ˆ (L) DS ˆ L G(L) χu s ) χu Gχv s ˆ ( χ0 G  EE N # ˆ (L+1) χ y s + χv G(L+1) DS ˆ (L+1) χ y s ) ˆ L+1 G ( χv G N =: A1 + A2 + A3 + A4 .

(13.22)

Here we have argued that the above bound holds for arbitrary fixed values of the θˆn . Thus it also holds after the average over these variables, denoted by Eˆ , is taken. Of the four terms A1 , . . . , A4 , found by expanding the two sums in (13.22), we will now find bounds for A1 , the one most easily handled, and A4 , the most complicated one. Corresponding bounds for the two mixed terms A2 and A3 can then be found by “interpolating” the provided arguments. Let E(. . . |J ) denote the conditional expectation with respect to the σ -field generated by the family {θk }k∈J . Due to independence this means that  E(X|J ) =

 ...

X

<

τ (θn )dθn .

(13.23)

n∈J

ˆ (L) and G ˆ (L+1) are independent of the variables The re-sampled resolvents G (θn )n∈J . Thus, by the rule E(X) = E(E(X|J ))

(13.24)

for conditional expectations, " # ˆ (L) χu s χu Gχv s χv G ˆ (L+1) χ y s ˆ χ0 G A1 := EE " # ˆ (L) χu s E( χu Gχv s |J ) χv G ˆ (L+1) χ y s ˆ χ0 G = EE  CE( χ0 G(L) χu s )E( χv G(L+1) χ y s ).

(13.25)

In the last estimate we have used the bound E( χu Gχv s |J )  C, e.g. Theorem 3.1, that the distribution of (θˆn )n∈J is identical to the distribution of (θn )n∈J , and that χ0 G(L) χu and χv G(L+1) χ y are stochastically independent. This bound for A1 is of the form required in Proposition 13.1.

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We continue with A4 , where we use (13.24) again and then apply Hölder’s inequality to the conditional expectation: " # ˆ (L) DS ˆ (L+1) χ y s ˆ L G(L) χu s χu Gχv s χv G(L+1) DS ˆ L+1 G ˆ χ0 G A4 := EE N N

ˆ E( . . . s . . . s . . . s |J ) = EE " ˆ (L) DS ˆ L G(L) χu 3s |J )1/3 ˆ E( χ0 G  EE N × E( χu Gχv 3s |J )1/3

# ˆ (L+1) χ y 3s |J )1/3 . ˆ L+1 G × E( χv G(L+1) DS N

(13.26)

We will now bound the three conditional expectations on the right separately. As s < 1/3, we can use Theorem 3.1 to bound the second factor by E( χu Gχv 3s |J )1/3  C < ∞

(13.27)

uniformly in (θk )k∈J and z. ˆ to get To bound the first factor, we start from the definition of D  ˆ (L) DS ˆ (L) P 3s P SL G(L) χu 3s , (13.28) ˆ L G(L) χu 3s  C

χ0 G

χ0 G N N ∈J ∩ L

ˆ (L) P = 0 for  ∈  L . The unitary diagonal operator Dω note here that χ0 G commutes with P and thus L (L) −1 L L

P S χu = P S N G N (Dω S N − z) χu

L −1 L = P Dω S N (Dω S N − z) χu

= P χu + zP G(L) χu

 1 + |z| P G(L) χu .

(13.29)

ˆ (L) does not depend on the variables (θn )n∈J . Thus, The re-sampled resolvent G using Theorem 3.1 again, we get ˆ (L) DS ˆ L G(L) χu 3s |J ) E( χ0 G N  ˆ (L) P 3s E((1 + |z| P G(L) χu )3s |J ) C

χ0 G ∈J ∩ L

C



ˆ (L) P 3s .

χ0 G

(13.30)

∈J ∩ L

In a similar way we find ˆ (L+1) χ y 3s |J )  C ˆ L+1 G E( χv G(L+1) DS N



 ˆ (L+1) χ y 3s .

P S NL+1 G

∈J ∩cL+1

(13.31)

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 ˆ (L+1) χ y = P χ y + By a calculation as in (13.29) we have P S NL+1 G ˆ (L+1) χ y = |z| P G ˆ (L+1) χ y . We conclude zP G  ˆ (L+1) χ y 3s |J )  C ˆ (L+1) χ y 3s . (13.32) ˆ L+1 G E( χv G(L+1) DS

P G N ∈J ∩cL+1

Combining the bounds (13.27), (13.30) and (13.32) into (13.26) we arrive at  ˆ (L) P s P G ˆ (L+1) χ y s ) ˆ ( χ0 G A4  C EE ∈J ∩ L ,  ∈J ∩cL+1

=C





E( χ0 G(L) P s )

∈J ∩ L

E( P G(L+1) χ y s ). (13.33)

 ∈J ∩cL+1

Here  it was also used that J has a fixed finite number of elements and s 1/3 ˆ thus ( j∈J x3s )  C j∈J x j . The last identity uses that (θn ) and (θn ) are j (L) (L+1)  identically distributed and that χ0 G P and P G χ y are stochastically independent. The bounds (13.25), (13.33) and related bounds for the mixed terms A2 and A3 combine via (13.22) and (13.16) to prove Proposition 13.1.   13.3 The Start of an Iteration We plan to use (13.13) as the first step in an iterative argument, where the next step consists of applying (13.13) again, but this time with E( χv G(L+1) χ y s ) on the left hand side with v  as the new origin. However, before doing this we need to replace G(L+1) with the original G, which can be done by reasoning similar to the decoupling argument of the previous section. Proposition 13.2 For every s ∈ (0, 1/3) there exists a constant C = C(s) < ∞ such that   E( χ0 Gχ y s )  CLd−1 E( χ0 G(L) χu s ) E( χx Gχ y s ) |x |∈{L+1,L+2}

|u|=L

(13.34) uniformly in z with 1 < |z| < 2, L ∈ N and y ∈ Zd with |y|  L + 2. Proof According to Theorem 13.1 we need a bound for E( χv G(L+1) χ y s ) in terms of fractional moments of the full Green function G for each fixed v  with

v  ∞ = L + 2. We start from the resolvent identity G(L+1) = G + G(L+1) T (L+1) G

(13.35)

and expand T (L+1) =

 (w,w  )∈∂

χw T (L+1) χw . L+1

(13.36)

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Combining both yields E( χv G(L+1) χ y s )  E( χv Gχ y s )  +C E( χv G(L+1) χw s χw Gχ y s ).

(13.37)

(w,w  )∈∂ L+1

With the goal of factorizing the expectation on the right we fix (w, w  ) ∈ ∂ L+1 and re-sample over the variables θn for n ∈ J˜ := Cv ∪ Cw ∪ Cw . With independent random variables (θ˜n )n∈J˜ independent from the θn , but with identical distribution, define  ˜ ˜ := D (e−iθn − e−iθn )Pn , (13.38) n∈J˜

˜ Dω,ω˜ := Dω − D,

(13.39)

˜ U ω,ω˜ := Dω,ω˜ S = U ω − DS,

(13.40)

˜ := (U ω,ω˜ − z)−1 . G

(13.41)

The resolvent identity ˜ − G DS ˜ ˜ G G=G

(13.42)

yields E( χv G(L+1) χw s χw Gχ y s )

˜ y s ) ˜ ( χv G(L+1) χw s χw Gχ  EE ˜ y s ) ˜ Gχ ˜ ( χv G(L+1) χw s χw G DS + EE =: B1 + B2 ,

(13.43)

where E˜ denotes averaging over the variables θ˜n . Also writing E(. . . |J˜ ) for the conditional expectation with respect to the σ -field generated by the variables (θn )n∈J˜ and arguing as in the previous section, one has 7 8 ˜ y s ˜ B1 = EE E( χv G(L+1) χw s |J˜ ) χw Gχ  CE( χw Gχ y s )

(13.44)

Hölder’s inequality yields

7 ˜ B2  EE E( χv G(L+1) χw 2s |J˜ )1/2 8 ˜ y 2s |J˜ )1/2 . ˜ Gχ × E( χw G DS

We have E( χv G(L+1) χw 2s |J˜ )  C and, by an argument as above,  ˜ y 2s |J˜ )  C ˜ y 2s . ˜ Gχ E( χw G DS

P Gχ ∈J˜

(13.45)

(13.46)

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This leads to the bound

⎛ ˜ ⎝ B2  CEE



⎞ ˜ y s ⎠

P Gχ

∈J˜

C



E( χx Gχ y s ).

(13.47)

x : Cx ⊂J˜

Collecting (13.43), (13.44) and (13.47) into (13.37), using that ∂ L+1 has CLd−1 elements, and ultimately applying Proposition 13.1 completes the proof of Proposition 13.2.  

14 Proof of Band Edge Localization We finally have reached the point where all the main results of the previous three sections can be put together to prove Theorem 3.5. Specifically, we will use the Combes-Thomas-type bound of Proposition 11.1, the Lifshits tail estimate of Proposition 12.2 and the decoupling estimate in the form provided in Proposition 13.2. Also frequently enter will the a priori boundedness of fractional moments established in Theorem 3.1. We will show the following fact which is equivalent to Theorem 3.5 (now in the normalization introduced at the beginning of Section 12 and only stated for the upper band edge): For 0 < s < 1/3 there exist δ > 0, α > 0 and C < ∞ such that E( χ0 G(z)χ y s )  Ce−α|y|

(14.1)

for all y ∈ Zd and all z ∈ C with 1/2 < |z| < 2, |z| = 1 and arg z ∈ [dλ0 − δ, dλ0 ]. Here χ0 and χ y are the characteristic functions of the cubes C0 and C y introduced in Section 13.1. Making the z-dependence explicit we write G(z) = (U ω − z)−1 and G(L) (z) = (U ωL − z)−1 in this section. The bound (14.1) implies E( χx G(z)χ y s )  Ce−μ x−y ∞ for arbitrary x, y ∈ Zd due to ergodicity. It suffices to prove (14.1) only for those z which in addition satisfy |z| > 1. To see this, use the identity G∗ (z) = −U ω (U ω − z¯ −1 )/z¯ = −U ω G(1/z¯ )/z¯ ,

(14.2)

which implies (14.3)

χx G(z)χ y = χ y G∗ (z)χx = χ y U G(1/z¯ )χx /|z|.  Inserting the partition y χ y and using that χ y Uχ y = 0 for |y − y| > 1 we conclude 1 

χ y G(1/z¯ )χx . (14.4)

χx G(z)χ y  |z| |y −y|1

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This shows that (14.1) carries over to z with 1/2 < |z| < 1 once it has been proven for 1 < |z| < 2, which is assumed for the remainder of this section. Proposition 12.2 shows that the probability that U Lk has an eigenvalue close to eidλ0 is small for the sequence Lk found there. Combined with the Combes-Thomas bound Proposition 11.1 this can be used to show smallness of the fractional moments E( χ0 G(Lk ) (z)χu s ) on the right hand side of (13.13) for values of z close to eidλ0 . Proposition 14.1 For any s ∈ (0, 1/3) there exist a sequence of integers Lk with Lk → ∞, g > 0 and C < ∞ such that d/(d+2)

E( χ0 G(Lk ) (z)χu s )  Ce−gLk

(14.5)

for all k sufficiently large, any z ∈ C such that 1 < |z| < 2 and arg z ∈ [dλ0 − −2/(2+d) , dλ0 ] and any u ∈ Zd with |u| = Lk . Lk Proof Let δ L > 0, to be specified later. The Combes-Thomas estimate Proposition 11.1 states that there exists B > 0 independent of L such that

χ0 G(L) (z)χu 

2 −BLδL e δL

(14.6)

for all z with dist(z, σ (U ωL )) > δ L and all u ∈ Zd with |u| = L. This takes care of the realizations ω such that the values of z are far enough from σ (U ωL ). The Lifshits tail estimate takes care of the realizations where this is not the case, in the sense that such instances are very unlikely. We set  

  G = ω | dist z, σ U ωL > δ L and  

  (14.7) B = CG = ω | dist z, σ U ωL  δ L . Making use of (14.6), we can write by means of Hölder’s inequality       E χ0 G(L) (z)χu s = E χ0 G(L) (z)χu s 1{ω∈ G } + E χ0 G(L) (z)χu s 1{ω∈ B } 

 2s −sBLδL  e E 1{ω∈ G } δ sL    1/t   1/t + E χ0 G(L) (z)χu st E 1{ω∈ B } , 

with 1 < t < 1/s and 1/t + 1/t = 1. Since E(1{ω∈ } ) = P( ) E( χ0 G(L) (z)χu st )  C for st < 1, by Theorem 3.1, we get

(14.8) and

       1/t e−sBLδL E χ0 G(L) (z)χu s  C + C P dist z, σ U L  δ L . (14.9) s δL To be useful for our purpose, this last quantity need to decay as L → ∞, which requires Lδ L → ∞. On the other hand, we need δ L → 0 for the probability that z is a distance δ L only away from σ (U L ) to be very small, for suitable z in a neighborhood of the band edge eidλ0 . In particular, this holds for the choice δ L = 1/Lβ and any β ∈ (0, 1).

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More specifically, with this choice of δ L Proposition 12.2 yields the existence of a sequence Lk with Lk → ∞ and positive γ¯ and C such that d(1−β/2)/t − γ¯ Lkdβ/2 t

βs

1−β

E( χ0 G(Lk ) (z)χu s )  Ce−sBLk Lk + CLk

e

(14.10)

β

for all k, 1 < |z| < 2, dλ0 − 1/Lk  arg z  dλ0 and |u| = Lk . The choice β = 2/(2 + d) leads to equal exponents of Lk in the two exponentials on the right hand side of (14.10). Choosing g < min(sB, γ¯ /t ) and requiring k to be sufficiently large we can absorb the power terms in (14.10) into the exponentials and arrive at (14.5)   We proceed with the proof of (14.1) by fixing s ∈ (0, 1/3) and choosing the −2/(2+d) sequence Lk and g as in Proposition 14.1. We now also choose δk = Lk . Proposition 13.2 says that  E( χ0 G(z)χ y s )  CLd−1 E( χ0 G(Lk ) (z)χu s ) k |u|=Lk



×

|x |∈{L

E( χx G(z)χ y s )

(14.11)

k +1,Lk +2}

if |y|  Lk + 2. Let 1 < |z| < 2 with arg z ∈ [dλ0 − δk , dλ0 ]. This along with Proposition 14.1 imply, for k sufficiently large,  2/(2+d) E( χ0 G(z)χ y s )  CL2(d−1) e−gLk E( χx G(z)χ y s ). (14.12) k |x |∈{Lk +1,Lk +2}

With the constant C from (14.12), fix L = Lk for k sufficiently large such that b := CL2(d−1) e−gL

2/(2+d)

#{x ∈ Zd : L + 1  |x |  L + 2} < 1

(14.13)

and get from (14.12) that E( χ0 G(z)χ y s )  b

max

x ∞ ∈{L+1,L+2}

E( χx G(z)χ y s ).

(14.14)

Note that E( χx G(z)χ y s ) = E( χ0 G(z)χ y−x s ), which allows to iterate (14.14). If x , x(2) , x(3) , . . . is one of the chains of sites obtained in this way, then the iteration may be continued as long as |x( j) − y|  L + 2, i.e. at least |y| − 1 times. For the last entry in the chain we use Theorem 3.1 to bound L+2 ˜ In (14.14) this leads to the bound E( χx( j) G(z)χ y s ) by C. ˜ log b |y| ˜ L+2 −1 = C e L+2 |y| . E( χ0 G(z)χ y s )  Cb b Thus we have proven (14.1) with C =

C˜ b

and α =

(14.15)

| log b | . L+2

Acknowledgements E. Hamza would like to acknowledge support through a Junior Research Fellowship at the Erwin Schrödinger Institute in Vienna, where part of this work was done. Also, E. Hamza and G. Stolz would like to express their gratitude for hospitality at the Institut Fourier of Université de Grenoble during visits at crucial stages of this work.

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